Is there a reasonable and studied concept of reduction between regular languages?
Barrington, Compton, Straubing and Thérien showed, in their paper Regular languages in $\mathsf{NC^1}$, that if the syntactic monoid of a regular language contains a nonsolvable finite group then the language is $\mathsf{NC^1}$-complete with respect to $\mathsf{AC^0}$-reductions (these are reductions computed by polynomial size, constant depth circuits with unbounded fan-in). Barrington's theorem implies that all regular languages are in $\mathsf{NC^1}$, and so such regular languages are complete for the set of regular languages under $\mathsf{AC^0}$-reductions.

Since we know that $\mathsf{AC^0} \neq \mathsf{NC^1}$ (for example, the parity function is in the latter but not in the former), regular languages in $\mathsf{AC^0}$ cannot be complete. For example, the language $a^*b^*$ isn't complete. Similarly, $\mathsf{AC^0}[p] \neq \mathsf{NC^1}$, showing that the language $(aa)^*$ isn't complete.

The simplest example of a language which satisfies the condition above is the language of all words over $S_5$ (the symmetric group on 5 elements) which multiply to the identity. The syntactic monoid of this language is $S_5$, which is a nonsolvable finite group. The slightly smaller alternating group $A_5$ would also work.