As discussed in the entry for √AR, for a considerable time in Tolkien’s life the basis for the word “and” was the root √AR with the sense “beside”, so that Q. A ar B “A and B” originally had the sense “A beside B”. However, at some point during the writing of the Lord of the Rings he decided that the Sindarin word for “and” was a, making √AR no longer suitable for its etymology.
From this point forward Tolkien toyed with two possible roots for “beside; and”, either √AD and √AS, with another option √ÑAR considered and rejected in 1957 (PE17/169). It seems Tolkien vacillated between the √AD and √AS, so an exact timeline is hard to nail down. Their primary difference would be in the prevocalic form of Sindarin “and”: either edhil adh edain [ada > aða] or edhil ah edain [asa > aha] for “elves and men”. The most detailed breakdown of these two possibilities appeared in Tolkien’s notes on words in The Lord of the Rings, probably written in the late 1950s (PE17/41). In these notes he kept flipping back and forth between ancient asa and ada, though ultimately settling on ada.
However, ah appeared in the title of the document Athrabeth Finrod ah Andreth “The Debate of Finrod and Andreth” most likely written in 1959 (MR/329), and in a 1968 note Tolkien said the primitive form was as with S. ah “and” before vowels and a before consonants (VT43/30). So either Tolkien reversed himself again and adopted √AS, or he continued to vacillate. For purposes of Neo-Eldarin, I think it is best to assume the root was √AS.
One result of the change of √AR >> √AS/√AD is that the Sindarin prefix ar- could no longer mean “beside” as it did in Noldorin. Indeed, in notes on The Rivers and Beacon-hills of Gondor written in the late 1960s he said “Arnen originally was intended to mean ‘beside the water’, sc. Anduin, but ar- in this sense is Quenya, not Sindarin” (VT43/17). This leaves us with no good word for “beside” in Sindarin; at one point I coined a neologism sa for this purpose, but it is a real stretch.
As a final note, these 1950s and 1960s roots were not the first time Tolkien used √AS for something like “beside”. All the way back in the 1910s, Tolkien had the root √AS(A) in both the Quenya and Gnomish Lexicons (QL/33; GL/48) with derived forms like ᴱQ. ar “to, against, next, on (wall)” (QL/33), G. hath- “close to, by, beside, touching” (GL/48), and [maybe] G. art “beside, alongside of” (GL/20), though the last form may be unconnected given the unlikeliness of s > r in Gnomish.
A root appearing in several sets of etymological notes from 1957 glossed “warmth” and forming the basis for Q. árë “sunlight, warmth” (PE17/148). This is further supported by the archaic form †ázë for this word (LotR/1123), with intervocalic [s] > [z] > [r] as usual. The form Âs was also mentioned as an ancient name for the Sun in some Silmarillion notes (MR/380). At one point Tolkien connected this root to S. Asfaloth which he translated as “Sunlit Foam” (PE17/18); he then rejected this derivation saying the root form was actually √GAS, but since the ancient form was given as √AS in several other places, perhaps he changed his mind.