The word ᴱN. muig “cat” appeared in Early Noldorin Word-lists of the 1920s (PE13/150). In the Gnomish Lexicon of the 1910s, Tolkien instead had G. {mio >>} miog “cat”, along with masculine and feminine variants miaug or miog “tom cat” and {miauli >>} miaulin “she-cat”. The derivation of these early forms isn’t clear.
Neo-Sindarin: I would retain ᴺS. muig “cat” for purposes of Neo-Sindarin, derived from ᴹ√MIW “whine” via an abnormal vocalization: m(i)wikē > mu̯ike > muig. Elaran pointed out that a more normal development would be ✱mŷg < miukē.
A word for “crumb” in Early Noldorin Word-lists derived from primitive ᴱ✶smeigé (PE13/150), illustrating how ei sometimes became ui in Early Noldorin (HPITN/§4.1.5). It might be connected to similar words like ᴱN. mig “little” (PE13/125).