Tolkien used a variety of similar roots for Elvish words having to do with “weaving”, many of them tied to the name of Q. Vairë. The earliest of these was a pair of roots ᴱ√GWERE “whirl, twirl, twist” and unglossed {ᴱ√WIÐI >>} ᴱ√GWIÐI in the Qenya Lexicon of the 1910s, the latter with derivatives like ᴱQ. ’winda “woof” and ᴱQ. ’windele “loom” (QL/103-104). The connection between ᴱ√GWERE and weaving is more obvious in the contemporaneous Gnomish Lexicon where Tolkien said that ✶gwer- “wind, turn, bend” was often used as “plait or weave”, much like ✶gwidh- (GL/46). The most notable weaving word derived from 1910s ✶gwer- was G. Gwerlum “Gloomweaver” (GL/46).
Nothing of this blended paradigm remained in The Etymologies of the 1930s, where Tolkien instead had ᴹ√WEY “wind, weave” as the basis for ᴹQ. Vaire/N. Gwîr “Weaver”, and in this document the root was blended with ᴹ√WAY “enfold” in Quenya because wei > wai (Ety/WEY). Tolkien seems to have abandoned this phonetic rule by the time he wrote his Quenya Notes (QN) from 1957, where he instead said:
> Owing to the use of gwae / gwaew “wind” as in Gwaehir, we must have √WAYA = blow, or be disturbed. √WAYA cannot therefore be used = “weave”, and Vairë has no connexion with winds or stories. EITHER Vairë must become name of Osse’s wife: Q váya is used of sea (as waters, motion). OR Vairë’s name be changed: sc. to Vérë, √WER- “twine, weave”, were-, weave (PE17/33).
Ultimately Tolkien made neither of these changes to Vairë, and this section was rejected. In a set of roots from December 1959 (D59) Tolkien said “√WIRI, weave; hence Vaire (literally ‘weaving’), not from WAY” (PE17/191). The name Q. Vairë “Ever-weaving” was also derived from √WIR in notes associated with the Quendi and Eldar essay from 1959-60 (VT39/10). However, in notes from the late 1960s Tolkien said “weaving with cross-threads or withes was represented by the distinct base {WAY >>} WIG, often in strengthened form waig-” (VT42/10 and VT42/29 note #27). So it seems Tolkien continued to vacillate on the weaving roots.
Neo-Eldarin: For purposes of Neo-Eldarin, I think the 1959-60 root form √WIR lets us salvage the largest number of words, and it is not clear how Vairë would be derived from late-60s waig-.
A variant of √LED in notes from the late 1950s or early 1960s that Tolkien considered to explain the derivation of S. lembas “waybread” (PE17/60). The root ᴹ√LEN “(?road), way” also appeared earlier in The Etymologies of the 1930s as the basis for N. lembas (EtyAC/LEN). Tolkien may have settled on this root when he revised the earlier root ᴹ√LED (Ety/LED) “go, fare, travel” >> √DEL “walk, go, proceed, travel” in the 1959-60 Quendi and Eldar essay (WJ/360), deciding that its inversion √LED was used mainly in Quenya (WJ/363).
A possible precursor to this root is ᴹ√LĒ “go, fare” in the Declension of Nouns (DN) from the early 1930s with derivative ᴹQ. lesto “journey” (PE21/12). This is turn is probably a later iteration of ᴱ√LEHE “come, be sent, approach” from the Qenya Lexicon of the 1910s, given as a variant of ᴱ√ELE (QL/52). It is probably the basis for contemporaneous G. len “come, arrived” and G. lentha- “come towards speaker, approach, draw near” (GL/53) and possibly ᴱQ. lehe- or ᴱQ. lehta- “ride” from the English-Qenya Dictionary of the 1920s (PE15/76).
For a discussion of other parallel roots, see √LED and √DEL.