If boy was born to a Jewish father and a non-Jewish mother and later converted in his life, would he obliged to call himself "ben Avraham" when called up for an aliyah, or would he be allowed to take his father's name?
This question it is similar to the case of a person who is adopted by a Jewish family and can call himself after the name of his adoptive father. When an adopted child is called to the torah, how are they named? has my answer to that question It might not be an exact duplicate because of the specific case given. In that answer, I state that I have known people who are known by the name of the adoptive father.
Given the general case, we can see from the other answers to this topic, that the Jewish father and the nonJewish son have no halachic connection whatsoever. The answer as far as aperson who converts to Judaism and is adopted by a Jew is the same no matter who the adoptive father may be. In any case, as explained in the other answers, a person can take the name of his adoptive father.
The only difference that would occur, is that if the adoptive father is a Levi (as an example from an actual case) then he is not called with the term "Levi" in the name because he is a yisrael.
As an example if the adopted son is named Moshe and his adoptive father is Shmuel HaLevi (made up names) the he would be called to the Torah as Moshe ben Shmuel.
In the case of the question, it would depend if the boy had converted and became part of the family of the biological father or not. In that case, it would be like an adoption. Otherwise it might be considered an embarrassment for the father to have his name used. If both parties consider it an honor, then it is acceptable.
An interesting point is found in masecta Yevamos 101B (Art Scroll 101B4 bottom of the page) in which Rav Shmuel Ben Yehudah explains that he is a ger. Thus we see that even as far back as the gemara a ger could use the name of someone else as the patronymic.
Beit Yosef, Even HaEzer 129:20, quoting Moses ben Isaac
מצאתי שצריך שיכתוב בלשון דלישתמע מיניה שהוא גר כגון שיכתוב הגר או בן אבינו אברהם ―
I found that one needs to write in language that implies that the person is a convert, for example write ―hager, the convert, or son of Avraham Avinu
(emphasis added). This was regarding a ger who had written a get and discussed the question as to what name needed to be used for the document to be valid.