What formal methods and criteria could be used to evaluate the message of the bible?

From a traditional Protestant point of view, according to the Bible, our highest authority is the Bible. It must be because "all Scripture is given by inspiration of God" (2 Timothy 3:16), and who or what can be of a higher authority than God?

It is self-authenticating to those whom God calls.

There must be a humble willingness to give the Bible a fair hearing. Faith is by hearing, and effectual hearing is by the word of God (Romans 10:17).

Tell him the Gospel anyway because the Gospel is the power of God unto salvation of all who believe (Romans 1:16). External evidence is not the power of God.

Read to him a portion of John ch 3. At the root of all intellectual pretences is a sinner who loves darkness rather than light because his deeds are evil (John 3:18-21). Sinners love the benefits of sin, they have been bribed: how can any judge in a court of law be dispassionate in weighing the evidence when he himself has a huge vested interest in the case?

Tell him gently and kindly that he is hell-bound because he is under sin, even dead in sin, that God became a man to take the punishment of his people and to live perfectly righteously on their behalf, so that all who repent and believe in the Lord Jesus Christ are delivered from the curse of God's law, saved from sin and hell, and given God's Holy Spirit to enable them to live for God.
Tell him his proud unwillingness to read the Bible for himself, and to hang onto his sins, will do him no good, and land him in the eternal fires of hell.

If any attempt is made to give an authority outside of the Bible to prove the Bible to be true then that authority is clearly being presented as a higher authority and better evidence than the Bible.

Or put it this way: if you were to give your friend evidence that the Bible is true, and that evidence came not from the Bible but from some other source, then you are asking him to put his confidence, not in the Bible, but in your other source.

"Let God be true, though every man a liar"....If a very honest man gives his testimony in court, its a bit rich to say "We will not believe him until his evidence is corroborated by a bunch of liars".

Tell your friend there is no evidence, there cannot be any evidence above the Bible, because the Bible is given by inspiration of God (2 Timothy 3:16,17); it is God's word, and how can there be any evidence better than the word of God?

To try to offer evidence better than God's evidence is deeply insulting to God. To ask for evidence better than that of God is likewise deeply insulting to God.

What he is really saying is "Don't quote the Bible at me, don't tell me Bible truths, because I hate the light". You must kindly tell him Bible truths, because those are what he needs.

If he will not believe the message of the Bible, hopefully by reading it for himself, then nothing else will do.

Having said all, one piece of internal evidence that the Gospel is true is that it is all in agreement with the Old Testament Scriptures, all "according to the Scriptures" (1 Corinthians 15:2,3). The Jews who today mostly continue to reject Christ, accept the Old Testament was all written before the time of Christ, yet those Scriptures ....( eg Isaiah 53, "They have pierced my hands and my feet; I can count all my bones, they look and stare at me" Psalm 22:16-18; and scores of others)... clearly prove Christ is the promised Messiah of the OT, the Saviour of the World.

But, and finally, it will take far, far more than mere evidence to persuade him; if he is asked he may well admit that:

And when he had looked round about on them with anger, being grieved for the hardness of their hearts, he saith unto the man, Stretch forth thine hand. And he stretched it out: and his hand was restored whole as the other.

And the Pharisees went forth, and straightway took counsel with the Herodians against him, how they might destroy him. (Mark 3:5,6)

What is needed is not evidence that it is true but the Word of God itself:

For the word of God is quick, and powerful, and sharper than any twoedged sword, piercing even to the dividing asunder of soul and spirit, and of the joints and marrow, and is a discerner of the thoughts and intents of the heart.

Neither is there any creature that is not manifest in his sight: but all things are naked and opened unto the eyes of him with whom we have to do. (Hebrews 4:12,13)

After all, he needs to be persuaded not just that the Bible is from God, but that God is loving, kind, merciful, honest, true, reliable, a God of peace and goodness. All his prejudices about God need to be removed before he will receive salvation. He needs the Gospel of Christ.

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 15:3-8 (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.

Creation

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 Church

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, as he graciously did for me.