When $G$ is finite, for arbitrary $H$ you can consider the quotient $G/H$ as an association scheme in the sense of [Zieschang, Paul-Hermann. An algebraic approach to association schemes. Lecture Notes in Mathematics, 1628. Springer-Verlag, Berlin, 1996. xii+189 pp. ISBN: 3-540-61400-1 MR1439253].
The notion of association scheme generalizes (at least...) that of group, that of distance regular graph, and that of (some types of?) building. When $H$ is normal, then the association scheme $G/H$ is the same thing as the association scheme associated to the group $G/H$, so in a sense, for $H$ non-normal considering the association scheme $G/H$ is a quite natural.
(In the infinite case, you can probably do more or less the same... I don't recall having seen infinite association schemes, though)
(Of course, this does not answer your question... but as Theo pointed in an earlier answer, I do not think $G/H$ is a groupoid in any sensible way in general, so the bit of structure I am mentioning might be a useful consolation prize!)