This root was the basis for Elvish words for stars and starlight, especially in the Sindarin branch of the Elvish languages. It first appeared as ᴱ√Gil- in the Gnomish Lexicon of the 1910s with derivatives like G. gil- “gleam”, G. giltha “white metal, †silver”, and (probably) G. gail “star” (GL/37-38). Its Early Qenya derivatives were ᴱQ. ilsa “mystic name of silver” (QL/42) and ᴱQ. īle “star” (GL/37), indicating the true form of the root was probably ᴱ√ƷILI, since initial voiced stops were unvoiced in Early Qenya, so that ancient ✱gīle would become ✱✱kíle, not íle.
This root appeared as ᴹ√GIL “shine (white or pale)” in The Etymologies of the 1930s with derivatives like N. geil “star” and ᴹQ. Ilma “Starlight” (Ety/GIL). The root regularly appeared in this unstrengthened form in Tolkien’s later writing, for example as √GIL “shine (white)” in a 1955 letter to David Masson (PE17/152) or as gil “white or silver light” in a 1958 letter to Rhona Beare (Let/278). In one place Tolkien considered giving it a kil- variant, analogous to √GAL vs. √KAL for “(golden) light” (PE17/50), but that seems to have been a transient idea. In other notes dating to the late 1950s Tolkien gave the root in strengthened form as √NGIL “silver glint” (MR/388; PE17/22) and also √GIL >> √ÑGIL as the basis for the initial element of the name S. Gil-galad (PE17/23).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to assume the root was originally unstrengthened √GIL, but was sometimes strengthened to √ÑGIL to produce words like Q. ñille “silver glint”.