Somewhere in my travels, I read that Jesus can be considered as Adam reincarnated. Is that a possibility? "First Adam flops, second Adam triumphs" - is that an idea some of you know?
No. Jesus may not be considered Adam re-incarnate.
Yet it's not hard to figure out where somebody might have gotten that idea. This is just a case of not understanding the terminology being used. Somebody got some of the words cross-wired¹ without understanding the concept.
In Christianity Jesus is known as the "Second Adam" or "Last Adam" but the naming has nothing to do with reincarnation! The title refers to a role rather than a personal ID. As far as personas go they are actually quite different. Adam was a man like you or me. He died and stayed dead. Jesus was God himself who had taken on human form. God and man at the same time.
Jesus not only did what neither Adam nor of his descendants could do (live in perfect obedience to God) but by taking on himself Adam's guilt he also reversed the process Adam started and provided a way for those who had rebelled (Adam and all us after him) to be reconciled back to God.
When Christians talk about a "Second Adam" it is not a reincarnation being talked about, but this principle of reversing the curse brought by the "First Adam".
¹ This could be intentional or accidental. History is full of deliberate miss-understandings of Christianity (e.g. the accusation of cannibalism in reference to communion) but I suspect this is simply a case of lack of knowledge. Rather that actually researching what is properly meant by various expressions, it is a natural tendency to run with them out of context. This tendency can not only result in very bad doctrine, it is the source of most of the worlds conspiracy theories.
There's nothing in scripture that supports the idea of reincarnation, meaning that the soul of a deceased person begins anew in the body of another.
Jesus is referred to as a "new Adam" here, which may be what you heard:
Though this is plainly read as an allegory, especially within the context of the whole chapter, meaning that both Adam and Jesus are the beginnings of a new kind of human being. The first Adam bringing forth man crushed under sin, and the second "Adam" bringing forth man given a new life through faith in Christ. There isn't any indication anywhere that Adam was reincarnated into Jesus.
In fact a man/woman (and their soul) is described as something that lives only once.
The Church Fathers referred to Jesus as the New Adam
So the idea has been around a long time. Reincarnated is definitely not the right term for being a "new anything". Jesus is "Incarnate of the Holy Spirit" as it says in the Nicene Creed. Which means His human flesh was created in the womb of the Virgin Mary by the power of the Holy Spirit, not reincarnated. I think reincarnation is a Hindu dogma, definitely not one found in Christianity.
Wow. The answer is, of course, yes, Jesus can be considered Adam reincarnated. This idea originated in the writings of Edgar Cayce. I wouldn't necessarily expect EC to be embraced by modern Christians.
To Jesus's credit, He did not focus on reincarnation as a philosophical truth. There is great advantage to this, because even from a context within reincarnation, we each have One eternal life. The here and now is here, and now, and nowhere else is important, not your past lives, not mine.
That said, it is written "Elijah came and they knew him not. And His disciples knew He was speaking of John the Baptist." I believe this is Matthew 12:17, though I don't have a Bible handy. EC cites this as evidence of reincarnation in the scriptures.
Jesus as first-begotten of the Father refers to the human Archetype. Jesus is within you, and within me. Or you can deduce from transitivity if you like: "Whatsoever you do to the least of my brothers, you do it to Me." (And through transitivity, Jesus, being within your brother, is within you.)
The first-begotten in time, as viewed historically, would be Adam. So, if you are viewing Adam as the human archetype, the first human, you are viewing Christ.
I would not ask questions such as these on the Internet, with all respect. You're going to get heady answers as you've received already. Reincarnation is thoroughly rejected in the West, because a fundamentalist approach wants to make God a certain way. God is not what God is because we want Him to be so (thank God!).
This is more understood in the East. God is the way God is, not because an American with an M-16 huffed and puffed that God is so. Though I will soften a bit and say that in the West, reincarnation is rejected because it seems to make God cruel. If life is difficult, why must we relive it? Yet Life as made by God is not the way we perceive it (thank God again).
You are better of asking God Himself what the Truth is about things. Or, ask Jesus. If I am right and Jesus is within you, then Jesus is available to you. A 2,000 year old book suffers from 2,000-year-old misunderstandings, as you can plainly see from the other answers.