I used to be in the same position as your friend, and it took me many years to take the claims of the Bible seriously.
At the time, I was doing a degree in Philosophy of Science, which discusses the nature of scientific evidence. One of the key principles is the Duhem-Quine Thesis, which states that it is impossible to test a hypothesis in isolation. That is, you cannot directly test the truth, only the effects of the truth.
As I began to explore the Bible, I found I engaged with it in a similar way. The Bible presents a certain world view, and I needed to ask myself whether the effects of that world view was what I saw around me.
In light of this, I would say the writers of the Bible put forward three pieces of evidence for the truth of the gospel (or, at least, these are the three that helped me).
Historical eye-witness of the ressurection
The Apostle Paul spends the first half of 1 Corinthians 15 providing the historic, eye-witness evidence for the ressurection:
For what I received I passed on to you as of first importance: that Christ died for our sins according to the Scriptures, that he was buried, that he was raised on the third day according to the Scriptures, and that he appeared to Cephas, and then to the Twelve. After that, he appeared to more than five hundred of the brothers and sisters at the same time, most of whom are still living, though some have fallen asleep. Then he appeared to James, then to all the apostles, and last of all he appeared to me also, as to one abnormally born. 1 Cortinthians 5:3-8 (NIV)
And at the end of his discussion, stresses the absolute importance of this evidence:
And if Christ has not been raised, your faith is futile; you are still in your sins. 1 Corinthians 15:17 (NIV)
The Apostle John also places great weight on eye-witness testimony:
Jesus performed many other signs in the presence of his disciples, which are not recorded in this book. But these are written that you may believe that Jesus is the Messiah, the Son of God, and that by believing you may have life in his name. John 20:30-31 (NIV)
As does Luke:
With this in mind, since I myself have carefully investigated everything from the beginning, I too decided to write an orderly account for you, most excellent Theophilus, so that you may know the certainty of the things you have been taught. Luke 1:3-4 (NIV)
Jesus also teaches this point in Matthew 16, when the Pharisees demand evidence beyond what has already been given:
The Pharisees and Sadducees came to Jesus and tested him by asking him to show them a sign from heaven. He replied, “When evening comes, you say, ‘It will be fair weather, for the sky is red,’ and in the morning, ‘Today it will be stormy, for the sky is red and overcast.’ You know how to interpret the appearance of the sky, but you cannot interpret the signs of the times Matthew 16:1-3 (NIV)
Jesus discusses this incident with his disciples a moment later:
Do you still not understand? Don’t you remember the five loaves for the five thousand, and how many basketfuls you gathered? Or the seven loaves for the four thousand, and how many basketfuls you gathered? Matthew 16:9-10 (NIV)
My understanding of this incident is that Jesus is saying the miracles he has already performed, and have been witnessed by large crowds, are enough evidence. The Pharisees shouldn't need to demand more.
The Apostle Pauls says that
For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse. Romans 1:20 (NIV)
Creation itself ought to speak on God's behalf. This is both physical nature, like beautiful sunrises and incredible music, since it was made by God. But it also include human nature, like our compassion and sense of justice, since we were made in God's image.
Personally, I would add our observation of suffering and sin into this - and this topic was actually the things that brought me to Jesus. The Bible explains the negative parts of human nature as well as the positive parts, and I believe it explains it far better than any other source I've found. The way in which humans are in constant tension between incredible good and incredible evil can only be adequately explained by the Bible. I found Paul's description of himself very compelling:
I do not understand what I do. For what I want to do I do not do, but what I hate I do...For I have the desire to do what is good, but I cannot carry it out. For I do not do the good I want to do, but the evil I do not want to do—this I keep on doing...So I find this law at work: Although I want to do good, evil is right there with me. Romans 7:15,18,19,21 (NIV)
The Apostles barely every stop talking about how the church itself is one the primary pieces of evidence for the truth of the gospel:
If I am delayed, you will know how people ought to conduct themselves in God’s household, which is the church of the living God, the pillar and foundation of the truth 1 Timothy 15 (NIV)
Watch your life and doctrine closely. Persevere in them, because if you do, you will save both yourself and your hearers. 1 Timothy 4:16 (NIV)
In everything set them an example by doing what is good. In your teaching show integrity, seriousness and soundness of speech that cannot be condemned, so that those who oppose you may be ashamed because they have nothing bad to say about us. Titus 2:6-8 (NIV)
...so that in every way they will make the teaching about God our Savior attractive. Titus 2:10 (NIV)
Live such good lives among the pagans that, though they accuse you of doing wrong, they may see your good deeds and glorify God on the day he visits us. 1 Peter 2:12 (NIV)
There should be something about the way the church treats people, both inside and outside the church, that shows people the Bible's teaching is true.
Of course, as other have pointed out, ultimately this isn't a logical decision, but a spiritual one. We must pray that God chooses to open the eye's of your friend, and he graciously did for me.