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” probably from around 1968 (VT47/26 note #26), 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.