First, the facts: As a Pharisee, Nicodemus could undoubtedly read and write. Nicodemus maintained a secret interest in Jesus' teachings and ministry that he was not able to share openly with his colleagues on the Sanhedrin. Nicodemus found himself in that lonely position for almost all the years of Jesus' ministry. (He most likely met Joseph of Arimathaea late in that period.) Nicodemus had access to writing materials and time on his hands for writing. And Nicodemus was in Jerusalem every time Jesus came to town. He may not have been able to stay very close to the Lord, but definitely had a sharp interest in hearing him speak.
One last fact: most of John's Gospel takes place in Jerusalem.
Now, the speculation: It is possible the Pharisee made notes in the hours or days after his night time meetings with Jesus. It is possible he wrote accounts of Jesus' public appearances also. It is further possible Nicodemus was so desperate for someone to share things with that he reflected on paper simply as a coping method with practically no thoughts of posterity. In any of these cases, it is highly likely such a unique personal journal would have survived (40 years?) until the beloved disciple began to compile his sources and compose his account.
There is more to consider: At best, Nicodemus was no better than the 2nd ranking Pharisee on the Council, and Gamaliel stayed out of politics (as, evidently, did his mentor Hillel). The Sadducees in power shot Nicodemus down pretty quickly, and the fact he saw Jesus at night means he was risking his position, so it seems pretty clear he had no peer to turn to.
Jerusalem's Pharisees probably had the inside track on picking up cheap paypyrus or parchment from local merchants. So Nicodemus was undoubtedly able to afford a modest amount of personal 'office supplies' as part of a Pharisee's general lifestyle. At the very least, he would have had enough parchment to make the customary phylacteries.
So Nicodemus absolutely had the ability to keep a journal of his thoughts about Jesus, if he decided to. Nicodemus absolutely had the leisure time to keep a journal, if he wanted to. And Nicodemus absolutely could afford the expense of the hobby, if he chose to.
As a scholar, as one who studied the law, as one who liked to ask questions and ponder things, as one who enjoyed reading... Nicodemus would naturally have wished he could remember Jesus' captivating, mysterious words more clearly. However long it took, and however much he wrote, the odds that he wrote something could actually be greater than not... especially on the additional chances he did feel a burden for posterity.
I'm not saying he necessarily remembered and put down word for word what Jesus said. And I'm not saying whatever Nicodemus wrote was copied into John's Gospel word for word, either. Even if I wanted to say that, there's no basis for arguing it. (Also, see my last post.) But I do think Nicodemus most likely wrote something and the beloved disciple most likely saw it.
If we accept that he was a historical figure in Jesus' life, this seems highly likely.
What else we might do with this suggestion, if anything, is a whole other topic...
[Update: I hope it is clear that these hypothetical writings of Nicodemus actually have nothing whatsoever to do with the thinking behind my next post on Johnine Historiography. That post stands alone, regardless of this one.]