The dichotomy started when some Christians (notably, evangelicals) began to define "religion" along the lines of this one:
Religion is a set of rules that people must follow in order to obtain God’s blessing.
Since Christians are saved by grace (and not by works), Christianity would not be a religion according to this definition. Source: Dictionary of Christianese.
The rest of this answer attempts to trace possible origins of the re-definition of religion which in turn produces this dichotomy. Disclosure: Key insights in this answer (Karl Barth, John Stott, metonymy) I obtained from Dr. Randal Rauser's Christian Post newspaper opinion article which is worth reading word by word.
Dichotomy as competing definitions
"Christianity as religion" has been contrasted not only with "Christianity as relationship" but also with "Christianity as the gospel" or with "Christianity as Christ". The common motivation is clearly to zoom into the essence of Christianity by excluding other elements of religion, as James A Fowler, a pastor and PhD in Biblical Studies, attempted to do in his 1998 book Christianity is NOT a religion shown by the table of contents (pdf here). The book started with this epigram:
Christianity has mistakenly been defined and described as a religion in which morality and belief-system in correspondence to the Book are regarded as the basis of the role-playing and problem-solving of the Christian life.
Christianity is Christ!
and then delve into these chapters: Christianity is Not Religion, Christianity is Not a Book-Religion, Christianity is Not Morality, Christianity is Not a Belief-System, Christianity is Not Epistemology, etc.
At the end of the day, it's a matter of definition as this blog article and another article argued the reverse position of how historically (as the OP pointed out) people have been referring to Christianity as a "religion". Both articles in fact advocate that we should refrain from spreading "Christianity is not a religion" because by doing so we will lose the status of Christianity as "the one true religion", or "the religion of religions".
Because it's a matter of definition that no official institution seems to promote, the problem of tracing the origin of the dichotomy will be similar to tracing how certain idioms became popular. So this becomes a problem of etymology, and the tools to find the origin will be similar to what lexicographers use when building an etymological dictionary.
Dichotomy usage in the wild
While doing research for this answer, I came across the Dictionary of Christianese - The casual slang of the Christian church... authoritatively defined. I think the phrase "Christianity is a relationship not a religion" can indeed be seen as a "Christianese" (delightfully constructed word, BTW), so naturally the dictionary has an entry for it, prefaced by a blog-article style introduction which like the 2 articles above is also arguing that it's a matter of definition. True to its name, the official entry has usage citations from published sources (journals, books, encyclopedias, etc.) starting from 1958 as the earliest citation.
1958 Episcopal Church, Diocese of Fond du Lac J. of the Annual Convention 86 : May we go forward remembering that Christianity is not a religion which merely lays upon us weak, human beings the hopeless task of living an impossibly good life helped only by the example of a man who lived a perfect human life 2,000 years ago, but rather that Christianity is a relationship to God whereby He communicates to us His strength and vitality which enables us to live on a higher plane.
It also includes a citation from Billy Graham as well (as the OP suspected):
1971 Graham The Jesus Generation 148 : [Consider] the novel thought that Christianity was not so much a “religion” as a relationship with a Person.
Dichotomy origin in specific definitions of religion
Can we do better in finding the earlier origin of the dichotomy, maybe not so much in the explicit use of the phrase "Christianity is a relationship not a religion" but a thought, a concept, a distinction, or an idea that prominent theologians used to construct a thesis related to "religion" which in turn lend itself to the dichotomy? So this becomes a task that Bible background / theological dictionaries tackle, such as articles describing what Religion / Law / Repentance / Messiah / After-life meant to various groups of people around the time of Jesus. The articles typically will also address a cluster of ideas around those concepts, citing key documents, movements, or people that could have contributed. Therefore we are looking for theologians with widespread influence who could have contributed to defining "religion" in a way that makes it contrary to "relationship".
Karl Barth as a possible origin
Dr. Randal Rauser, a professor of Historical Theology writing the Aug 2019 opinion article I mentioned above, offered an implied definition of "religion" which could serve as the opposite of "relationship" in the dichotomy, and thought that Karl Barth used that definition in his 1918 Romans commentary:
The speaker, in this case, is clearly not adhering to a conventional dictionary definition of “religion”. Instead, she is invoking a rhetorical use of the word, one popular among some Christians. It seems to me that it can be defined roughly like this:
religion. n. Those beliefs one holds and actions one undertakes to address the human problem and relate rightly to ultimate reality which are inconsistent with the revealed beliefs and practices commended by God and which seek to redress that human problem primarily through human effort.
This usage was famously invoked by Karl Barth in his much-lauded theological reflection on the book of Romans in which he “translated” Paul’s reference to the Law as “religion”. Thus, in Barth’s calculus, the law/religion are what we do to relate rightly to God while Jesus Christ is what God does to relate rightly to us.
For those who want to dig deeper into Karl Barth's contribution to this understanding of "religion", I found these 3 helpful resources which can serve as a starting point:
John Stott as a popularizer
The same Christian Post opinion article plus others I found on the net points to popular quotes attributed to John Stott, like the following, which seems to be taken from a book / interview (unfortunately I cannot find the definitive origin of the quote in John Stott's writings, so I don't know the date):
“I’ve learnt very early on that Christianity is not a religion and it is not an institution, it is a person. It was enormously helpful for me to discover that Christianity is Christ and and that what matters is a personal relationship to Christ.
So the verses which came to mean much to me were Philippians 3 verses 8,9, where the Apostle Paul says, “That what was gain to me I counted loss for Christ. Indeed I have counted all things but loss in comparison with the overwhelming gain of knowing Christ Jesus my Lord for whom I’ve suffered the loss of all things (Paul went on) and count them as refuse or rubbish in order that I may gain Christ and be found in him, not having my own righteousness based on Law, but the righteous that comes from God through faith in Christ.”
It’s all Christ, Christ, Christ, Christ. Knowing Him, loving Him, serving Him, trusting Him, gaining Him. ‘To me to live is Christ’, Paul said, and I think, I hope without boasting I can say the same, it is a Person.”
But I did find this quote from his 1992 book The Contemporary Christian, recipient of Christianity Today 1994 Critics Choice Award. The book was recently (June 2019) republished by IVP in 5 smaller books, the first being The Gospel - A Life-changing Message, and I found what looks like a publisher excerpt here. As you can see, IVP even prominently started the summary of the book as follows:
What is the authentic gospel? How do we answer the sceptics?
Christianity is not a religion, but God's good news for the world. This implies that it has both a divine origin and a human relevance: it comes from God and it speaks to our condition.
Here's the first paragraph of the book's introduction to the first part (The Gospel), which was already present back in 1992:
Christianity is not a religion, let alone one religion among many. It
is God’s good news for the world. The Christian gospel has both a
divine origin (it comes from God) and a human relevance (it speaks
to our condition). So, before we ask the question, ‘What is the gospel?’,
we must explore the logically prior question, ‘What is a human
Conclusion: dichotomy origin as an effect of the evangelistic ethos
Now that we have identified several prominent personalities influencing the community which promoted a particular meaning of "religion", the next step is to inquire the motivation of that change, since ultimately language is a human product. From the survey of usage it does not look like it is a temporary change; the change seems to be gaining force toward permanence, at least within certain Christian groups. The Wikipedia article on semantic change includes a list of types as well as motivations of semantic change. In the article, Dr. Rauser pegged the type of change as metonymy. How about motivation? Below is my personal argument that the motivation is the fervent desire to spread the gospel, which resulted in a semantic change to better fit the group's ethos.
Regardless of the terms used to describe Christianity, it is a unique "religion" which breaks a lot of attempts to categorize it. It is also an intensely proselytizing "religion" because Jesus, the founder, was Himself providing the imperative to bring God's good news (the new covenant) to the end of the world in order to gather all sheep into his Kingdom at the end of time. Who can beat God when He Himself sets His will to call us to Him?
All the persons quoted in the articles and books above were "on fire" from within to communicate this good news, and have to face a multitude of challenges from a variety of audiences who have "prejudiced" themselves against this good news by trying to put the Good News in a box, a category that will make this Good News "safe", that will put them "in control" by barricading themselves in a comfort zone of their own making. Thus an academic makes his/her own notion of religion using high-sounding scholarly language to recast the good news as a respectable "systematic theology" that he/she can attend to whenever comfortable, at the cost of a real encounter with God. A non-intellectually inclined person falsely thinks of God as a being to be "appeased" by prayer & fasting & sacraments done mostly in a behavioral / ritualistic manner, also at the cost of a real encounter with God. In both cases, the Good News would not have produced the effects that the real God intended within the person.
But God's way to save a person (through the Holy Spirit) is to initiate the action from above, literally pursuing the target like a hunter, and demolish any categories that become an idol / barrier to the full effect of the good news, which includes the full personal revelation of God's being to a potential convert's heart. It is sometimes necessary for the Holy Spirit to reconfigure this person's mind and ideas so this full personal revelation can touch the person's heart. It is then no wonder that many committed Christians including heavy-weights like Karl Barth, Billy Graham, and John Stott felt free as ambassadors of God to adjust the long-standing dictionary definitions of "religion" when they feel that the definitions are obstructing the force of God's redemptive actions in human hearts & minds.
In the final analysis, it's not that the older definition of "religion" is bad or wrong; it's clear that dichotomy wouldn't work without a re-definition of "religion". The dichotomy also doesn't imply that theology, sacraments, and activities are in themselves bad; the dichotomy is a rhetorical device to put those things in perspective so the "relationship" aspect which implies Christ and the God-initiated Good News can truly shine.