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.
The root √TAR appeared unglossed in the Quendi and Eldar essay of 1959-60 as the basis for Q. tára “tall, high” (WJ/417), a word that elsewhere was derived from √TĀ/TAƷ “high” (Ety/TĀ; PE17/186). The root √TAR was glossed “stand” along with derivative tāra “tall” in rough notes on the back of a discussion of the comparitive from around 1967 (PE17/186). The past tense for Q. tarne “stood” appears in other notes from this period, along with Q. astarindo, artarindo or astarmo “bystander” (PE17/70-71). √TAR “stand” seems to be a later iteration of ᴹ√THAR “stand” from the Quenya Verbal System of the 1940s; of this earlier root Tolkien said it “is only used [in describing the location of things] — except, of course, with reference to persons or animals when they are noted especially as ‘standing’ (not sitting or lying) — of mountains, high hills, towers, pillars” (PE22/126).