How did the three wise men get their name?

Answer 1

The author of Matthew's Gospel did not refer to wise men, he referred to magi. The magi were priests of the Zoroastrian religion of Persia. The allusion to wise men has two bases: the first is the name of their god, Ahura Mazda ("Wise Lord") and the second is that the magi were widely respected throughout the Middle East for their learning and wisdom.

Initially, Matthew's intention was to show that even the priests of this great religion would want to worship Jesus. Eventually, the Zoroastrian connection became less important and most English-language translations began to translate magi simply as "wise men".

Answer 2

Mr. Harfield's answer is correct insofar as the "wise men" part of the answer goes, but the "three" part is also worth discussing, because The Bible never actually says there were three of them. It's implied that there were more than one, because the word "magi" is plural, but the exact number is never specified. The Bible does mention three gifts, and that's probably where the traditional number of magi came from.