The correct answer is “one of my friends”. You normally use the expression “one of” with a plural noun. It could be a group of people or things. “One of” informs the listener that you are talking about one member of the group. Since you are talking about only one member in the group, you use a singular verb after the plural noun.
Examples: One of my friends is going to America. *One of the books that I am planning to read is “Very Good Jeeves”.
You have many friends, but right now you are talking about only one of them. If you had only one friend, then you would have said, “My friend is going to America.” Similarly, the second example suggests that you are planning to read many books and one of them happens to be “Very Good Jeeves”. It is important to remember that the noun cannot immediately follow “one of”.