As far as the Gospel of John goes, the idea makes sense that Nicodemus was best known by that name in John's community. The Johannine community was probably located in Asia minor. Although it may have included Jews who fled Jerusalem after 70 c.e., John's audience was obviously Greek-speaking, since his gospel was written in Greek.
As far as the "historical Nicodemus" goes, we should be aware that the Pharisees were by no means limited to Judea and the Galilee, although this is where they are located when we read about them in the Gospels. We have evidence that the Pharisees themselves were often fluent in Greek, interacted freely with Greeks and used Greek names. The Jewish Encyclopedia states that "the Pharisees made the Torah a power for the education of the Jewish people all over the world, a power whose influence, in fact, was felt even outside of the Jewish race." Their means of doing so was the synagogue, a institution largely of their creation. Thus, when we hear in Acts about apostles speaking in numerous synagogues in the Greek-speaking cities, we can safely presume that Pharisees were often involved in their leadership.
In addition, we should consider that Pharisees often travelled (Matthew 23:15). Many of them, like Paul, grew up outside of Judea with their families in Greek-speaking cities. Hillel the Elder, the grandfather of Paul's mentor Gamaliel, was raised in Babylon, where Greek was one of the official languages at the time. He became famous for his willingness to teach Gentiles, probably speaking in Greek.
Paul himself, of course, was fluent in Greek and used a Greek name when writing to churches. But in Acts 13, he is still called "Saul" long after his conversion.
Now in the church at Antioch there were prophets and teachers,
Barnabas, Simeon... and Saul. While they were worshiping the Lord and
fasting, the Holy Spirit said, “Set apart for me Barnabas and Saul for
the work to which I have called them.” ... When they had gone
through the whole island as far as Paphos, they came upon a certain
magician, a Jewish false prophet, named Bar-Jesus... But Saul, who is
also called Paul, filled with the Holy Spirit, looked intently at him
and said, “You son of the devil, you enemy of all righteousness, full
of all deceit and villainy, will you not stop making crooked the
straight paths of the Lord?
Although they resisted Hellenization, the Pharisees would likely use Greek names when appropriate, just as the Pharisee Saul did when he signed his letters "Paulos," or when Joseph Ben Matthias, who lived around the time that the Gospel of John was composed, wrote as "Josephus."
Regarding the Talmudic Nicodemus, he is not called Buni but "Nakdimon," which is basically the same Greek name as Nicodemus. Taanit 19b states
The Sages taught: Once all the Jewish people ascended for the
pilgrimage Festival to Jerusalem and there was not enough water for
them to drink. Nakdimon ben Guryon, one of the wealthy citizens of
Jerusalem, went to a certain gentile officer [hegemon] and said to
him: 'Lend me twelve wells of water for the pilgrims, and I will give
back to you twelve wells of water. And if I do not give them to you, I
will give you twelve talents of silver. And the officer set him a time
limit for returning the water.'
So whatever his Hebrew name might have been, Nakdimon ben Guryon was accustomed to speaking to Gentiles and used a Greek name.
Thus the historical Nicodemus may have used both his Hebrew and his Greek name, as Saul/Paul did. The most likely reason he is called only "Nicodemus" in the Gospel of John is that John's audience was Greek speaking and the historical Nicodemus lived far away, two generations in past.
Further reading: Jewish First Names Through the Ages