The early existence of this root was indicated in the Gnomish Lexicon of the 1910s as (unglossed) ᴱ✶haw- where it was the basis of words like G. hau- “to lie” and G. haus “bed” (GL/48). It reappeared in The Etymologies of the 1930s as ᴹ√KHAW “rest, lie at ease”, this time with both Quenya and Noldorin derivatives such as ᴹQ. hauta- “cease, take a rest, stop” and N. haust “bed” (Ety/KHAW). In The Etymologies Tolkien also said this root influenced the meaning of N. hauð “mound” so that it came to mean “grave, tomb” as well (Ety/KHAG, KHAW).
The root reappeared again in revisions made to the Outline of Phonology (OP2) around 1959 with a new gloss “cover up, hide away, lay in store”, along with an extended root √KHAWAD “store, hoard” that continued to influence S. haudh “(funeral) mound” (PE19/91). The root √KHAW >> √KHAWA also appeared in a page of roots having to do with “large and small” of unclear date (but probably late 1950s or early 1960s), with the derivatives Q. höa “big, large” and Q. haura “huge” (PE17/115).
Neo-Eldarin: For purposes of Neo-Eldarin, I think the 1930s meaning “rest, lie at ease” is probably most useful, and I would limit the meaning “lay in store” to its extended root √KHAWAD. As for the meaning “big”, I would limit it to (Quenya-only) Q. höa.
This root was primarily used as the basis for Elvish words for summer and, via its extended root √LAYAK, freshness and greenness. Its earliest appearance was as ᴱ√LAẎA “be alive, flourish” in the Qenya Lexicon of the 1910s (QL/52) where the Ẏ indicated an ancient palatal spirant, [c] or [ɟ]. In this period it had derivatives like ᴱQ. laiqa/G. laib “green” and ᴱQ. laire/G. glair “meadow” (QL/52; GL/39, 52).
In The Etymologies of the 1930s ᴹ√LAYA was unglossed with a single derivative ᴹQ. laire “summer”, appearing in a marginal note next to more detailed ᴹ√LAYAK (EtyAC/LAYA). In this document, Tolkien gave the derivatives of ᴹ√LAYAK as ᴹQ. laiqa “green” vs. N. lhoeb “fresh” (Ety/LAYAK). The Ilkorin form laig “fresh, lively; keen, sharp” was said to be blended with ᴹ√LAIK “keen, sharp, acute” (Ety/LAIK), so “✱fresh” was probably the original meaning of ᴹ√LAYAK.
The root √LAY appeared quite regularly in Tolkien’s later writings, but was never glossed. It was mentioned in the second version of Tengwesta Qenderinwa from around 1950 as an example of a bi-consonantal root (PE18/97). It appeared in 1957 etymological notes as the basis for “summer” and “green”: the former still Q. lairë but the latter now Q. laika (PE17/145, 159). It was mentioned again in a 1958 letter to Rhona Beare as the basis for “summer” and ”green”, with the Sindarin form S. laeg having largely been replaced by S. calen “green” (Let/282).
Most likely this root retained the same basic sense as it had in the Qenya Lexicon, connected as it was to words having to do with freshness and liveliness.