Earlier name of Emyn Muil in Lord of the Rings drafts from the 1940s, glossed “Border Hills” (TI/313), also appearing as Emyn Rain (TI/268). Its seems to be a combination of the plural of amon “hill” and rhain “border”.
Noldorin
rhain
adjective. free
rhain
noun. border
rhain
noun. border
lhain
adjective. free, freed
rhein
noun. border
emyn rhain
place name. Border Hills
duil rain
place name. Border Hills
minas tirith
place name. Tower of Guard
edrein
noun. border
lhein
adjective. free, freed
The Etymologies of the 1930s had N. {rhein >>} rhain “free” derived from ᴹ✶(a)ranı̯ā under the root ᴹ√RAN “wander, stray”; it also had a deleted variant erein, presumably from aranı̯ā (EtyAC/RAN). See also S. rain “erratic wandering” for a later appearance of a similarly derived word.
Neo-Sindarin: Many Neo-Sindarin writers adapt this adjective as ᴺS. rain “free” for purposes of Neo-Sindarin, as suggested in HSD (HSD), since the unvoicing of initial r to rh was a feature of Noldorin of the 1930s but not Sindarin of the 1950s and 60s. I would use it as “free” in the sense “✱unconstrained”.