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.
A neologism for “nest” coined by Helge Fauskanger for his NQNT (NQNT), based on ᴹ✶khau̯-stā “rest-ing”.