The word Q. Teler was a long established word for an Elf in Tolkien’s writing, though it ultimately came to refer only to the third tribe, and had a variety of derivations over time. In the Qenya Lexicon of the 1910s ᴱQ. Teler “little elf” was not given a root, but its stem form teleř- implies derivation from ✱ᴱ√TELEÐ (QL/90), which is consistent with its Gnomish cognate G. Tilith (GL/70). In Early Noldorin Word-lists of the 1920s Tolkien derived ᴱQ. Teler and ᴱN. Tiledh from primitive ᴱ✶Teled- (PE13/154).
In the first version of Tengwesta Qenderinwa from the 1930s Tolkien initially gave the root ᴹ√TELED for the third Elven kindred (PE18/34), but later in the same document he gave ᴹ√TELES (PE18/61), which is the form he used in The Etymologies of the 1930s with the gloss “hindmost, tarrier” (Ety/TELES). In The Etymologies he explicitly contrasted ᴹ√TELES with ᴹ√KYEL “come to an end” (> ᴹQ. tyel-) with which it was sometimes blended in Quenya (Ety/KYEL, TELES). However, in the second version of Tengwesta Qenderinwa from around 1950 Tolkien gave {√TELES >>} √KYELES as the basis for the name of the third tribe, with Teler- being the Telerin variant.
In Common Eldarin: Verb Structure from the early 1950s Tolkien reversed himself, deciding √TEL meant “close, end, complete” at which point primitive teles “come at rear, end a line or series” was restored (PE22/135). In rough notes associated with Notes on Names (NN) from 1957, he switched to primitive Teler- as the basis for the tribe-name, as supported by Sindarin forms like S. †teleir (PE17/139), and in the Quendi and Eldar essay of 1959-60 he said Q. Teler was based on the old agental suffix ✶-rŏ (WJ/371), again pointing to primitive ✱teler-.
Neo-Eldarin: For purposes of Neo-Eldarin, I would assume that the tribal name Teleri was based on ancient ✱tele-rŏ as noted above, but I would also assume √TELES “hindmost, last in a series” remained valid to preserve relevant derivatives from The Etymologies.
This root and ones like it were used for Elvish words for “silver” throughout Tolkien’s life. The earliest iteration of the root began with T-, however, appearing as unglossed ᴱ√TELEPE in the Qenya Lexicon of the 1910s with derivatives like ᴱQ. telpe “silver” (QL/91). Even at this early stage, however, the Gnomish equivalent was G. celeb (GL/25), but the reason for the t/c variation isn’t clear. The closest explanation is that palatal consonants like [c] became [tʲ] in Qenya vs. [k] in Gnomish (compare ᴱQ. tyava- vs. caf- “taste” from ᴱ√TYAVA) but this doesn’t explain why the Qenya form has initial t- rather than ty-.
Elsewhere in the Elvish languages of the 1910s there seem to be etymological variations of [k] vs. [t], such as ᴱQ. kitya- vs. G. tisca- “tickle” (QL/47; GL/70) and ᴱQ. talqe vs. G. celc “glass” (QL/88; GL/25), so perhaps ᴱQ. telpe vs. G. celeb “silver” is another example of this. Another explanation appeared in Early Noldorin word lists from the 1920s, where the primitive form was ᴱ✶kelekwé which produced ᴱN. celeb as usual but the Qenya form was ᴱQ. telqe with “k = t by dissimilation” (PE13/140), presumably away from q.
In The Etymologies of the 1930s Tolkien had the root ᴹ√KYELEP with variant ᴹ√TELEP, producing N. celeb but ᴹQ. tyelpe or ᴹQ. telpe (Ety/KYELEP). But Tolkien revised this entry, marking ᴹ√TELEP as questionable and introducing the Telerin form ᴹT. telpe < ᴹ√KYELEP, concluding that ᴹQ. telpe must be a loan from Telerin. This finally put N. celeb vs. ᴹQ. telpe (borrowed from Telerin) on a solid phonological foundation. Tolkien seems to have stuck with this explanation, mentioning this borrowing from Telerin to Quenya several times in his later writings, with the proper but now archaic Quenya form being Q. †tyelpë (Let/426; PM/356; UT/266).