Tolkien used a variety of roots for Elvish words for “small”. One early root was ᴱ√PIKI with variants ᴱ√PINI and ᴱ√PĪ from the Qenya Lexicon of the 1910s, unglossed but with derivatives like ᴱQ. pínea “small” and ᴱQ. pinqe “slender, thin” (QL/73). It also had derivatives in the contemporaneous Gnomish Lexicon such as G. pinig “tiny, little” and G. pibin “small berry, haw” (GL/64).
The root reappeared in The Etymologies of the 1930s as unglossed {ᴹ√PEK >>} ᴹ√PIK with derivatives like N. pigen “tiny” and N. peg “small spot, dot” (Ety/PIK; EtyAC/PIK). Further evidence for this root can be seen in later words like Q. piki- or pitya “petty” (WJ/389) and Q. pik- “lessen, dwindle” (MC/223). A variant root √PEY appeared in a list of roots having to do with large and small, with a single derivative Q. pia “little” [< ✱peya], but it was immediately followed by the forms pikina, pinke, pitya which point back to √PIK (PE/117).
Further evidence of early forms ᴱ√PINI and ᴱ√PĪ can also be seen in Tolkien’s later writings. There is S. ✱pîn “little” in S. Cûl Bîn “Little Load” (RC/536), ✶pī {“small bird” >>} “small insect” (VT47/35), and T. pinke “little-one, baby” (VT48/6), though the last of these might be from √PIK. In any case it seems Tolkien continued to use all of √PI, √PIK and √PIN to form words for little things into the late 1960s.
In The Shibboleth of Fëanor from the late 1960s, Tolkien changed pitya to Q. nitya in the name Q. Nityafinwë “Little Finwë” (PM/353, 365 note #59), which may indicate a replacement of √PIK by √NIK, another root used regularly in Tolkien’s later writings for “small”. But I believe √PIK and √NIK may coexist with slightly different meanings: “tiny” vs. “small”; see the entry on √NIK for further discussion.
A root introduced by Tolkien in Definitive Linguistic Notes (DLN) from 1959 with the gloss “close eyes, blink, wink” to provide a new explanation for the flower name S. alfirin (originally “immortal = not mortal”) after he decided the prefix al- meant “well” rather than “not” (PE17/146). He derived the names Q. pirindë and S. pirin for “a flower that opened and shut quickly with any change of light”. The root ᴱ√PIRI also appeared (unglossed) in the Qenya Lexicon of the 1910s with derivatives like ᴱQ. piri- “spin, turn”, ᴱQ. pirin “thin rod, pin” and ᴱQ. piruke “swirl, twisting, pirouetting” (QL/74). Finally, the word Q. pirë “toe” appeared in notes associated with the 1955 version of the poem Q. Nieninquë, perhaps representing a third meaning for this root (PE16/96).
Neo-Eldarin: For purposes of Neo-Eldarin, I think the gloss “blink, wink” for √PIR is not a great fit, since it is a contrived etymology Tolkien invented for a word he clearly meant to be something different originally. Q. tihta- is probably a better known verb for “blink”. For “spin”, I prefer a Neo-Eldarin root ᴺ√KWIR adapted from early ᴱ√QIŘI. However, I think ᴺ√PIR might be retained as a Neo-Root with the sense “✱cylinder”; I think this works as an explanation of Q. pirë “toe”, and allows the retention of early words like ᴱQ. pirin “thin rod, pin” and ᴱQ. pirinumbe “cylinder”.