The Gnomish Lexicon of the 1910s had G. authmeg “midday, high noon” (GL/20) as a combination of G. auth “sunshine, noontide” and (probably) the early root ᴱ√MEKE having to do with the center (QL/60). It also had G. saidhon “noon” derived from primitive ᴱ✶sek’·ðana (GL/66), and thus G. seg [< *sek-] “middle point” combined with G. dana “day”.

Neo-Sindarin: For purposes of Neo-Sindarin, I would update these to ᴺS. nedhor “mid-day, noon” as combination of [N.] nedh- “mid-” and aur “day”.