I’m working under the assumption that Gil-Galad is a descendant of Finwe, and that he was born at some point after the Dagor Bragollach, possibly even after the Nirnaeth Anoediad. I’m also assuming the Noldor don’t believe in ruling queens, which is all but stated in the text.
I don’t know whose kid he is, but I know why he went down in history as Fingon’s. Let’s look at potential parents (all living members of the house of Finwe). [under a cut for length]