⋄ Narothal ’Firefoot’ (RS:345,347,351) [Gandalf’s horse] – N. tâl ’foot’ (TAL-)
⋄ Narosîr ’Redway’ (RS:433) – N. sîr ’river’ (SIR-)
⋄ Narodûm ’Red Vale’ (RS:433) – for dûm ’valley’ see the discussion of the other names of the ’Red Vale’ below
I have put these three words together because of the common initial element meaning ’red’ or ’fire’, doubtlessly related to the stem NAR1- ’flame, fire’ with the derivative narwā ’fiery red’ > N. narw, naru ’red’. Compare also Nargalad, Nardol below.
Variation between o and u is a common phenomenon in Noldorin/Sindarin. For instance, u becomes o if the following syllable contains a, e, o or ō , but not if this u comes from the consonant w, as it may be the case here (narwā), unless one might take naro- as analogical. But we may also be dealing with o from the vowel u, take a look at the following commentary:
-u- suffix frequent in Q[enya] after el, al, see Q[enya] Structure; cf. kelu, telu, smalu, etc. (VT46:8)
This also applies to Noldorin and Ilkorin, which further often shift u > o; see e.g. N./Ilk. celon ’river’ < kelu + n (KEL-) or Celos, derived from kelu- ’flow out swiftly’ + -sse, -ssa as in Q. kelussë ’freshet, water falling out swiftly from a rocky spring’ (UT:426). And -u- is perhaps also be favoured after r, compare e.g. Q. erume ’desert’ (ERE-), Eru ’the One’ (UT:305).
Another interpretation might be that the first element is the genitive of *nar ’flame’. Genitival inflections were lost in Noldorin/Sindarin with the final vowels, but may have been preserved medially. This may explain the lack of lenition of sîr and dûm, as well as the usage of the circumflex – the compounds being put together out of earlier expressions like *naro sīre > *naro-sîr, but it does not explain the spirant mutation in Narothal.
A third explanation could be that the first part is actually ?naron, an adjective similar to caron ’red’ below. The consonant -n then causes nasal mutation of tâl to -thal (although other examples show lenition instead). In the case of Narosîr it is then assimilated to -ss- > -s-, compare S. besain < ’bread-giver’ < *mbassaniē; Q. massánie (PM:404). But this is very shaky indeed and cannot explain Narodûm.
And finally a fourth possibility is an adjective of the form ?narod < narātā, which may explain both Narothal < *narautthal < *narāt-tal and Narodûm < *narod-dûm < *narāt-dûm, but not Narosîr, where one would rather expect *Narothîr, as in N. ethir (ET-, SIR-). But after all, one should not expect that Tolkien had to use exactly the same element in all of these names.