The parentage of Gil-galad is definitely one of the questions Tolkien left up in the air. So we have a few options, and I'm not sure which we will go with. It is likely that the decision will be made based on where/when/how we want him to debut in the TV show.
As a son of Fingon, Gil-galad would be born in Hithlum, and would likely be a child after the Battle of Sudden Flame (ie in Season 5). Fingon would then send him to live with Cirdan on the coast and keep him out of the war (ie, off screen). While this is the version that made it into the published Silmarillion, it is not the case that Tolkien's writings support this as any kind of definitive choice. This was Christopher Tolkien grasping at straws and trying to make the story work out neatly, which he acknowledges in HoME. Still, we could make the same choice for the same reason.
As a son of Orodreth, Gil-galad would be fully grown by the time we have significant stories happening in Nargothrond - Beren and Luthien, Turin, and the fall of Nargothrond. So, he would likely play a role in these stories, and somehow escape the destruction that kills the rest of his family. We would need to invent a storyline for him, and it likely wouldn't be overly flattering (ie, he'd fail to save anyone or prevent any disasters). And there is no way Orodreth could send him away for safety reasons while not also sending off Finduilas, unless there's a huge age gap or something, and Finduilas is a fullgrown elf, but Gil-galad is just a baby when Turin shows up. I...don't like that solution. It *is* possible that Gil-galad could respond to the warnings of Ulmo's messengers, and speak against Turin at the council. Tolkien writes Gwindor as Turin's opponent in that scene, but we could have room for two dissenting voices - Finduilas' former love and her brother. That would give Gil-galad a reason to go to Cirdan in response to Cirdan's messengers, getting his story back on track, and he could just happen to be away from home when the dragon attacks. Still..that could be very problematic.
Keep in mind that the name Ereinion = Scion of Kings. Thus, whatever heritage we give him has to live up to that name and reflect how he could have ended up inheriting the title of High King of the Noldor when he did. Also, we have to keep in mind the Doom of Mandos and keep him out of the Fëanorean branch of the family, as they have to stay the Dispossessed and can never get the High Kingship back (and yes, in the earliest version, Tolkien did have him as a Fëanorean, but Celebrimbor ended up with that role). Oh, and Celebrimbor is in Nargothrond, by the by.
Making Gil-galad son of Fingon, son of Fingolfin, son of Finwë certainly makes the inheritance clear and you can see why someone would call him scion of kings - his father, grandfather and great-grandfather were all High Kings of the Noldor. Obviously, Turgon takes the title of High King at the death of Fingon, so we would have to come up with an explanation of that (Do younger brothers inherit before sons? Or is Gil-galad still a minor at the time and therefore Turgon's High Kingship is more of a regency thing? If Turgon holds the crown in his own right, how does it get back to Gil-galad instead of going to Earendil?)
This is where it becomes painfully clear that we don't know the elven laws of inheritance. Sure, we could *assume* that it's male primogeniture. But that wasn't the rule in Numenor (where we are told it was oldest child, son or daughter). So, what if the immortal(-ish) elves have different rules? After all...there is no reason to think your son will outlive your younger brother in this culture, so a brother is a perfectly valid/safe heir.
Fëanor claims to be Finwë's heir, and no one really bothers correcting or challenging him. But *technically* at the time of Finwë's death, Fëanor was still banished from Tirion. He and his brother had reconciled, but the years of his term of banishment weren't up; Fingolfin was still ruling as king in Tirion. Fëanor is thus never really called High King of the Noldor. He never had a coronation ceremony, so it's all rather unofficial when he stirs everyone up into an impromptu rebellion to return to Middle Earth.
After his death, there is therefore a question - are the Noldor in need of identifying Finwë's heir, or Fëanor's heir? Because with the death of Fëanor and the absence of Finarfin, Fingolfin is quite clearly Finwë's heir and thus the rightful High King. But....obviously not all of the Noldor held that opinion. So is Fëanor's heir his younger half-brother Fingolfin or his eldest son Maedhros? Maedhros resolves all of these questions with the following pronouncement: he identifies Fingolfin as High King by virtue of his being "the eldest here of the House of Finwë, and not the least wise." It doesn't matter if others (ie, his younger brothers) disagree with him, because he is the only one who could have challenged Fingolfin for the leadership; his concession makes the Fëanoreans Dispossessed forever.
I bring this up, because good luck justifying Gil-galad son of Orodreth son of Angrod son of Finarfin son of Finwë as High King of the Noldor while *ANY* other princes of the Noldor survive. Especially since there is no indication that women are ineligible to inherit and thus Galadriel's existence throws a spanner in those works (as she clearly would acknowledge Gil-galad as High King personally). Likewise, Elrond son of Earendil son of Idril daughter of Turgon is still around and swearing allegiance to Gil-galad...who is inheriting the High Kingship from Turgon (somehow).
Just some food for thought. We will need to resolve the questions surrounding inheritance in Season 3, as we will likely get to that scene of Maedhros renouncing his own claim and recognizing Fingolfin in the Season finale. If not, that's the opening of Season 4. Gil-galad's parentage can likely be put off for a bit longer, as he isn't born yet. The most we have to do is make sure his father (whether Fingon or Orodreth or someone else) is married by the Feast of Reuniting early in Season 4, so it doesn't impinge on the Andreth storyline.