I believe that he was actually the son of Finrod Felagund. This is never actually stated, but he names himself Gildor Inglorion of the house of Finarfin.
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In the first edition of LR (Appendices) Finrod was still the name of the third son of Finwë, as in the Quenta Silmarillion , and his son was Felagund (in QS also named Inglor ); it was still not until the second edition of 1966 that Finrod son of Finwë became Finarfin, and his son Inglor Felagund became Finrod Felagund
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-Note 9; Ch. 10 (The Attack on Weathertop); The Return of the Shadow, HoME 6.
So if Finrod Felagund was once Inglor Felagund, wouldn't Inglorion mean
son of Inglor which is the son of the the first King of Nargothrod? Now you say, "But Felagund had no wife." Perhaps...
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... for indeed she whom he (Finrod Felagund) had loved was Amarië of the Vanyar...
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-Quenta Silmarillion, Ch. 15, Of the Noldor in Beleriand
Maybe he was married to Amarië, or perhaps they were lovers. For whatever reason she did not go into exile with him, I still believe that she bore him at least one son: Gildor.