A noun for “holder, socket, hasp, clasp, staple” in The Etymologies of the 1930s derived from primitive ᴹ✶takmā “thing for fixing” under the root ᴹ√TAK “fix, make fast” (Ety/TAK). Here it seems the k vocalized to i to produce the diphthong ai, which then became ae, while the post-vocalic m softened to v and then became w as it often did at the end of words: *takma > taima > taem(a) > taev > taew. The Sindarin developments would have been similar.

Based on its long list of glosses, it seems taew could be used of fasteners in general, both movable like a hasp or clasp, as well as fixed like a staple or socket, as opposed to its Quenya cognate ᴹQ. tangwa which could be used only if a movable “hasp, clasp”.