The word G. aidha “a nest” appeared in the Gnomish Lexicon of the 1910s derived from primitive ᴱ✶ai-idh (GL/17). Immediately above it, G. aivor was glossed “nest” but this gloss was crossed out (see that entry for discussion). Elsewhere in the Gnomish Lexicon, Tolkien gave G. end “birds nest” from primitive ᴱ✶aı̯·ind, but this entry was deleted (GL/32).
Neo-Sindarin: I’d adapt this word as ᴺS. aedh “nest” for purposes of Neo-Sindarin, related to S. aew “bird”, since the primitive diphthong ai becoming the ae in Sindarin. Its primitive form might be ✱ai̯[w]dē.
The words G. aivor and aivin appeared in the Gnomish Lexicon of the 1910s with the glosses “rookery, colony of birds” (GL/17). The gloss of aivor was initially “nest”, and the form aivin was originally aivar, but Tolkien edited the entry to remove the gloss “nest” and change {aivar >>} aivin.
Neo-Sindarin: I’d adapt this word as ᴺS. aemar “rookery, colony of birds”, related to S. aew “bird” with the second element being S. bâr (mb-) “home”.