Common people might not think about writing things down until the eyewitnesses are nearing their death. However, as I've said before, it only takes ONE GUY to be forward thinking enough to beat those commoners by several decades. However, he had to be more than literate. He also had to be literarily minded. Personally, I think that ONE GUY was Matthew the publican-scribe. Here's why:
Imagine you're Matthew the tax collector. You carry around some simple writing tools that you use for book keeping when you have to travel, which is often. When somebody pays you, you unroll your satchel and scroll, set your tools on their table, make a record of the transaction, and put everything away again. You'd keep a small jar of ink, and you'd know in which towns to buy more ink when you were running low. As a tax-collector, you are in small part a travelling scribe.
Now Jesus calls you. And you want to serve him. And writing is not only a skill, it's a part of your practiced profession. But one more important detail is... you're Jewish! You grew up hearing people read the scriptures out loud. You've spent many hours rolling and unrolling scrolls, just like you saw done in the synagogue every Sabbath, so writing isn't a mystery to you. In fact, writing is part of your daily mindset.
But there's more. In the synagogues, among other things, you had sometimes heard that God told Moses to write, God told Isaiah to write, God told Jeremiah and other prophets to write... and apparently some of them wrote without being told to. You see Jesus go into the Synagogues, and they read the scrolls, and at least one time you think, "They should hear what Jesus said the other day!"
To birth writing, literary capability needs to meet literary mindedness. One simple thought only had to cross Matthew's mind one time. He already had the skills and the tools. He just needed the spark. "I could write something!" And writing itself was abundantly extant.
By the time of Hillel, famous Rabbis had people who wrote down their sayings. Pharisees commonly put writings on their foreheads or sleeves, so common people could see it and want to know what it said. Herod's Temple had inscriptions written on it by those who donated materials. The average person might not see writing all day long, but they might see it daily. And pages of writings - for synagogue attenders - would be seen at least weekly.
Maybe Jesus himself made a remark one day that got Matthew to thinking about the potential. Maybe Matthew offered to sell his writing tools for food money, and someone suggested he keep them for future use instead. It doesn't matter how it happened - the point is that all of these possibilities are extremely plausible or common scenarios that could have nudged Matthew into thinking about recording some speeches and current events.
But what if it wasn't Matthew?
The odds of that thought coming at least once [at least to some writer] seem as certain as anything. And my whole point - again - is that the thought only had to come once. In a purely oral culture, maybe writing would never occur until the eyewitnesses were about to die. But in a partly literate culture? If the task only had to occur to one literate person? If it only took one guy to decide to write down what he knew? More likely than not, somebody would write.
For the sake of argument, let's pretend there were two hundred literate christians out of the thousands in Palestine. With those numbers, what are the odds that just one of them compiled an account? Those odds would be the inverse of 200 to 1. Those odds would be 200 in 1. That's two hundred chances for one guy to do it. Those are pretty good odds.
By the way, written testimony doesn't merely back up an eyewitness who might leave or die - it also backs up that witness' own memory while they're still around! Again, this only increases the odds that the earliest christians would not have waited so long to make written records about Jesus' ministry years.
So, after all that... my faithful blog reader... if YOU had to pick the most likely New Testament person to put down written records more early than others, who would you choose?
So far, I'm sticking with Matthew.
Do you have a better (specific) suggestion?