Tolkien used the root √KIM and similar roots for a wide variety of purposes throughout his life. Perhaps the earliest of these was the Early Qenya word ᴱQ. kim- “heed” from the Qenya Lexicon of the 1910s, but all of its Gnomish cognates begin with g-, as in G. gima- “hear” and G. gimri “hearkening, attention” (GL/38), so this early root was most likely ✱ᴱ√GIMI. In the Quenya Verbal System of the 1940s, Tolkien had ᴹ√KIM “light on, find, come by” and along with a verb ᴹQ. kim- of similar meaning (PE22/103, 108, 125) but most likely this meaning of the root was transferred to √KHIR and Q. hir- “find” (PE17/75).
Finally in notes from 1969 Tolkien gave √KIM “edge, brink of”, replacing deleted forms √KIL and √KILIM, with the root √KIL being reassigned the meaning “choose, select” (which seems to be the normal meaning of √KIL throughout the 1950s and 60s). Note that the early root ᴱ√KILI the Qenya Lexicon of the 1910s also meant “edge”, but somewhat interestly there is a Gnomish word G. cim “blade”, perhaps indicating that this vacillation between √KIM vs. √KIL for “edge” had a longer history in Tolkien’s conceptual development of the languages.
Neo-Eldarin: For purposes of Neo-Eldarin, I would recommend using √KIM = “edge” and √KIL = “choose”.
This root has a long history in Tolkien’s writing, but it evolved in meaning over Tolkien’s life. The earliest form of this root was ᴱ√KILI “edge” from the Qenya Lexicon of the 1910s, with derivatives like ᴱQ. kíla/G. cilm “edge, rim” and ᴱQ. kilme/G. cail “lip” (QL/46, GL/24, 26). In The Etymologies of the 1930s it became ᴹ√KIL “divide” with derivatives ᴹQ. kilya/N. cîl “cleft, pass between hills, gorge” (Ety/KIL). This sense seems to have been abandoned when Tolkien changed ᴹQ. Kalakilya “Pass of Light” (LR/173) to Q. Calacirya (LotR/377).
In etymological notes from 1969 Tolkien gave √KIL “choose, select” vs. √KIM “edge, brink of”, along with various derivatives of √KIL having to do with choices (PE22/149). One of these, Q. cilmë, appeared as an element in words from several essays on Elvish names from the 1950s and 60s: Q. Essecilmë “[Ceremony of] Name-choosing” (MR/214; PE21/84) and Q. kilmessi “self-names, names of personal choice” (PM/339). But it seems even in the 1969 note mentioned above that Tolkien was considering the earlier senses of the root, since he first gave √KIL/√KILIM as the basis for edge words (PE22/149 note #26).
The difference in sense is not that great between 1910s ᴱ√KILI “edge” >> 1930s ᴹ√KIL “separate” >> 1960s √KIL “choose”, so Tolkien’s conceptual shifts on the meaning of this root seem to be gradual, with him still considering the earlier meanings into the late 1960s.