The word Q. pé was the main Quenya word for “lip(s)” for all of Tolkien’s life, but its derivation evolved over time. The root first appeared as unglossed ᴱ√PĒ in the Qenya Lexicon of the 1910s with the derivative ᴱQ. pé “the two lips, the (closed) mouth” (QL/72). The contemporaneous Gnomish Lexicon compared this Qenya word to G. beg “chin” or “beard” (GL/22), hinting that the actual root might have been ✱ᴱ√BĒ. In the Declension of Nouns from the early 1930s, the primitive form was given as PĒ “lips, mouth” (PE21/1, 38). In The Etymologies of the 1930s the root became ᴹ√PEG “(?outer) mouth” with derivative ᴹQ. pé “mouth” (Ety/PEG; EtyAC/PEG).
In Common Eldarin: Noun Structure from the early 1950s the root is given as unglossed √PEÑ (PE21/71), and in notes associated with the Quendi and Eldar essay of 1959-60 the primitive form is given as peñe (VT39/11). In both cases it was an example of a primitive form that resulted in ancient monosyllabic nouns from weak consonant loss: √PEÑ > ✶pē. In notes from the late 1960s Tolkien again gave ✶pē in a list of primitive monosyllabic nouns, but said “of these all except pe, su had probably lost a consonant in Common Eldarin”, implying the original form was actually √PĒ. But in green-note revisions made in 1970 to Outline of Phonology Tolkien had:
> ñ disappeared prehistorically, so that words such as peñ were for Quenya long monosyllabic nouns with only an initial consonant: pē (PE19/102 and note #168).
Thus it seems the lost ñ in √PEÑ was restored, though it could also be a remnant of the earlier version of this sentence from the 1950s that gave both peñ and maʒ as examples of consonant-loss.
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to assume the root is √PEÑ.
This root has a long history in Tolkien’s development of the Elvish languages, but its exact meaning is hard to determine because Tolkien rarely translated the root itself. The earliest appearance of the root was as unglossed ᴱ√LIPI from the Qenya Lexicon of the 1910s where Tolkien indicated it might be a dialectical variant of ᴱ√LIQI “flow, water; clear, transparent”; it had derivatives like ᴱQ. lipte- “to drip”, ᴱQ. litl “a tiny drop”, and ᴱQ. limpe “elfwine” (QL/54). It also had derivatives in the contemporaneous Gnomish Lexicon such as G. lib “drop, gout”, G. lib- “to drip”, G. limp(elis) “the drink of the fairies” (GL/54). I think the most likely meaning of this early root was “✱drip, drop”.
In The Etymologies of the 1930s, the root for “drip” was ᴹ√LIB. ᴹ√LIP appeared, but it was unglossed and its only derivative was ᴹQ. limpe “wine” (Ety/LIB¹, LIP). Thus it seems Tolkien split up the root from the 1910s, though exactly how isn’t clear. The last known mention of this root is as √LIP “oil” (without any derivatives) in a currently unpublished set of notes from 1968 (VT44/15). Wynne, Smith, and Hostetter suggested this might be connect to a (rejected) name for Christ: Q. Elpino, perhaps meaning “✱anointed” (VT44/15). It is unclear whether this √LIP “oil” was connect to its earlier iterations from which limpe “wine” was derived, or if it is was a reemergence of a different root such as ᴱ√ILI “shine oily” as suggested by Wynne, Smith, and Hostetter (VT44/20 note #7).