The word númen or núme was the Quenya word for “west” for much of Tolkien’s life, and was also the name of tengwa #17 [5] (LotR/1123). It is a combination of √N(D)Ū “go down” and √MEN “way” (PE17/64; Ety/NDŪ, MEN), so originally meaning “going down” or “way of the setting sun” (Let/361) in the same way that rómen “east” is the direction of the rising sun. Q. númen is thus similar in origin to English “occident”. As a standalone word it is usually númen, but in compounds or with inflections it is frequently reduced to núme-. For the Elves, númen is considered the primary direction, since it points towards Aman where the Two Trees were (LotR/1123; VT49/8).
Conceptual Development: ᴱQ. nūme “west” dates all the way back to the Qenya Lexicon of the 1910s where it appeared under the early root ᴱ√NUHU “bow, bend down. stoop, sink” (QL/68). The form númen did not appear as an independent word until The Etymologies of the 1930s (Ety/MEN), and númen was already the name of tengwa #17 in the earliest version of notes on The Feanorian Alphabet (PE22/23).
The Sindarin word for “sunset”, more literally “going down”, also used to mean “West” (LotR/1116, 1123; PE17/64, 88, 121). It was based on the root √NDU “sink, go down” (PE17/64), derived from primitive ✶ṇdūnē like its Quenya counterpart Q. andúnë (Ety/NDŪ). The overlong vowel in the last syllable of annûn is unusual, and was due to the influence of related dûn “west” (LotR/1116 note #1). I believe annûn is most generally used for “sunset”, and when used for “West” it means “the West” as a location, as opposed to a direction which is dûn.
Conceptual Development: N. annûn first appeared in The Etymologies of the 1930s, already with the derivation given above but only glossed “west” (Ety/NDU).
There was a similar word {nún >>} ᴱN. nuin in Early Noldorin Wordlists and the Early Noldorin Dictionary of the 1920s, a noun form of the verb ᴱN. nuv- “to sink, set” (PE13/151, 161, 164). Tolkien first gave it as nún “sinking, going down” as a derivative of primitive ᴱ✶numne (PE13/151), but later gave it as nuin “sinking down, going down” from primitive ᴱ✶nubhin (PE13/164). In the second instance he specified it was both “n[oun] & inf[initive]”. In the first instance Tolkien said that “Nún is used = West”.
Another likely precursor was G. nûmin “the west, sinking” from the Gnomish Lexicon of the 1910s (GL/61) which in turn was probably based on the early root ᴱ√NUHU “bow, bend down; stoop, sink” as suggested by Christopher Tolkien (QL/68; LT1A/Númë).