This root was connected to hand words for all of Tolkien’s life. Its periodic shifts between √MAH and √MAƷ had more to do with Tolkien’s ongoing vacillation on the form of the ancient velar spirant (voiced vs. voiceless) in Primitive Elvish. The earliest appearance of this root was as ᴱ√MAHA “grasp” in the Qenya Lexicon of the 1910s, with derivatives like ᴱQ. má “hand”, ᴱQ. maqa “handy, skilled”, and ᴱQ. māra “mighty, power, doughty; (of things) good, useful” (QL/57). In the contemporaneous Gnomish Lexicon the root was maχā with derivatives G. mô “hand” and G. manc “grip, grasp, hold” (GL/55); G. mora “good” may have also been related (GL/57).
In The Etymologies of the 1930s it appeared as ᴹ√MAƷ “hand” along with a related root ᴹ√MAG “use, handle”, with ᴹQ. mára “useful, fit, good (of things)” derived from the latter (Ety/MAƷ). In Tolkien’s later writings, the root √MAG was further differentiated from √MAƷ, becoming the basis for “good, useful”; see the entry on √MAG for further details. As for ᴹ√MAƷ “hand”, it was also mentioned in the Outline of Phonetic Development from the 1930s (OP1: PE19/48) and Outline of Phonology from the early 1950s (OP2: PE19/100 note #154).
In revisions to OP2, Tolkien decided the primitive velar spirant was unvoiced, a weak χ [x] rather than voiced ʒ [ɣ] (PE19/69 note #3; 72-73 note #22; 74 note #33). From this point forward, Tolkien sometimes rendered this root as √MAH (PE21/70; VT39/11), but it continued to appear often as √MAƷ even into the late 1960s (VT47/18-19) reflecting ongoing vacillation on the nature of the primitive velar spirant. As for its meaning in later writings, in Definitive Linguistic Notes (DLN) from 1959 Tolkien said it meant “serve, be of use” as opposed to words for “well” and “good” derived from √MAY or √MAG (PE17/162), and in another section of the same notes it was defined as “handle, manage, control, wield” (PE17/163).
Neo-Eldarin: For purposes of Neo-Eldarin, I think the last meaning is best for √MAH or √MAƷ. I think words for “good” and “useful” are better assigned to √MAG, and “excellence” to √MAY, though Tolkien often intermingled the meanings and derivatives of these three roots.
This root was intertwined with the root for “hand” words: √MAH or √MAƷ. Indeed, in The Etymologies of the 1930s the roots ᴹ√MAG and ᴹ√MAƷ were variants of one another, with ᴹ√MAƷ glossed “hand” and ᴹ√MAG glossed “use, handle”, the latter with the derivatives like ᴹQ. mára/N. maer “useful, fit, good (of things)” and N. maen “skilled, clever” (Ety/MAƷ|MAG).
In later writings, however, the sense of this root shifted more towards “good”. In etymological notes from the late 1950s Tolkien still gave both √MAG or √MAƷ as the basis for hand words (PE17/161-2), but in Definitive Linguistic Notes (DLN) from 1959, Tolkien gave √MAGA the gloss “good (physically)” (PE17/149) or “to thrive, be in a good state” (PE17/162). In notes on Eldarin Hands, Fingers and Numerals from the late 1960s, Tolkien said the root for “hand” was √MAƷA, and there was a distinct (but possibly related) root √MAG, described as follows:
> MAGA, a stem meaning “good” — but without moral reference, except by implication: sc. it was not the opposite of “evil, wicked” but of “bad (damaged, imperfect, unfit, useless)”, and the adjectival stem derived, ✱magrā, meant “good for a purpose or function, as required or desired, useful, proper, fit” (VT47/6).
A similar distinction between √MAƷA “hand” and √MAGA “good, useful” appeared in 1968 notes (VT47/19). In Late Notes on Verb Structure (LVS) from 1969, Tolkien gave √MAGA the gloss “have, possess” to serve as the basis for the irregular verb Q. mai-, but this note was crossed through and I suspect it was a transient idea (PE22/148 and note #24). Thus, “good, useful” seems to be the best interpretation for the root √MAG.