An invertible root meaning “noble”, itself an extension of √AR (PE17/147). Given its widespread use, Tolkien did not define this root until quite late. Its first clear mention was in Notes on Names (NN) and Quenya Notes (QN), both from 1957. In NN, Tolkien introduced invertible √ARA/RĀ to give an explanation of the element rod in names like S. Rodon “Vala” and S. Finrod, where Tolkien said:
> S raud “tall, high, lofty, eminent”. √ARA, RĀ. Cf. Q aran “king”. Q arta “high” < árată. S raud < form (a)rā́tă (PE17/118).
Earlier in The Etymologies of the 1930s, the second element of N. Finrod was N. rhaud “metal” (Ety/RAUTĀ), whereas ᴹQ. haran/N. aran “king” was derived from ᴹ√ƷAR “have, hold” (Ety/ƷAR). As for the later root √ARA/RĀ, Tolkien gave a lengthier explanation in QN:
> √AR. ARI-, ARA/RĀ, ARAT/ARAN. Original sense probably seen in adverb/preposition ✱ara/ar/ra, beyond, further than. The stems ARI/ARAN/ARAT mean “good, excellent, noble” — differing from √MAN in stating that any one specimen is “good of its kind”, excels, without necessarily implying that others are bad or marred. Hence ari-, good as prefix, ar/ara/aran as prefixes of excellence especially in royal names, aran king. Q arta noble, arato a “noble”, ráta- excel, surpass. S aran, arod (arāta); or raud excellent, noble, eminent (PE17/147).
Thus all these roots, including the extensions √ARAN and √ARAT, were elaborations on the sense “beyond” from base root √AR. The simple root √AR itself had a lengthy conceptual history; see that entry for details. As for ARA/RĀ, the connection between these forms and nobility seems to be firmly established by the late 1950s and afterwards, with primitive element Ara- “noble” mentioned in The Shibboleth of Fëanor from 1968 (PM/344, 363 note #43), and numerous other mentions of derived forms in documents from this period.
The root √AR has a long and complex history in Tolkien’s writing. For many years, it was the basis for the word ar “and”. Its earliest precursor was the root ᴱ√ARA or ᴱ√ƷARA in the Qenya Lexicon variously glossed “spread, extend sideways” or “wide places” (QL/32). The Gnomish derivatives of this root such G. garw “sown field” (GL/38) vs. ᴱQ. arwa made it clear the true primitive form was √ƷARA (in Gnomish, ʒ- > g-). Some of the early derivatives of √ƷARA such as G. gar(th), ᴱQ. arda “place” were later transferred to the root ᴹ√GAR so they could retain this gar-/ar- distinction.
Of the derivations that remained under √AR, the most notable were ᴱQ. are “beside, along” and the conjunction ᴱQ. ar(a) “but” (QL/32). The latter changed in meaning to ar “and” by the end of the 1920s, for example in the Oilima Markirya poem. This carried into the 1930s paradigm for the root ᴹ√AR, as seen by its entry in The Etymologies with its derivatives ᴹQ. ara “outside, beside” (the basic sense of the root) and ᴹQ. ar “and” (Ety/AR²). The most common Noldorin word for “and” in this period was likewise ar (TAI/150; SD/128-129), and in prefixal form ar- “outside, beside” sometimes developed a privative sense “without”, most notably in arnediad (†arnoediad) “without reckoning, numberless” as in N. Nirnaith Arnediad “(Battle of) Unnumbered Tears” (Ety/AR², NOT) which in Sindarin became Nirnaeth Arnoediad.
In some etymological notes from the 1950s Tolkien retained the root form ara “alongside” (VT43/33), but there were already cracks forming in this system, forced by Tolkien’s decision that the Sindarin word for “and” was a rather than ar, a change that first appeared Lord of the Rings drafts from the 1940s (TI/182). By the late 1950s Tolkien was experimenting with new roots √AD(A) and √AS for the meaning “beside” and the derivation of Q. ar, S. a “and”; see those entries for later developments in this semantic space of “beside”.
As for the root √AR itself, it shifted in meaning to “beyond, further than” in Quenya Notes (QN) from 1957, becoming the basis for “royal” roots like √ARAN “king” or √ARAT “noble” (PE17/147). In this revised meaning, it might still be able to retain a “privative” sense in Sindarin words like †arnoediad “unnumbered” (perhaps = “✱beyond numbering”), though it is also possible Tolkien simply never revisited the etymology of this Sindarin word.