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](http://www.ams.org/mathscinet-getitem?mr=MR1439253)]. 

The notion of association scheme generalizes (at least...) that of group, that of [distance regular graph](http://en.wikipedia.org/wiki/Distance-regular_graph), and that of (some types of?) [building](http://en.wikipedia.org/wiki/Bruhat-Tits_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!)