I'm afraid I don't know what this specific informal use of do is called. The regular use of do as a substitute, in "do I?" and "I do not", is I believe called do-support.
I think your case is similar to "could you give her this document too?" — "will do", which does sound American to my ears, perhaps southern U.S. or something.
What might also be relevant is why we omit some words but not others in ellipsis, and how we make our choice: to "are you coming to the party?, we could answer simply "no", or "no, I'm not", or "no, I am not coming". Or even "I might", "I might come" etc.
My guess would be that this usage of "might do" and "will do" is not old, though I am not sure.
Modals have always differed from
ordinary verbs in Germanic, and in the
course of the history of English, they
have diverged from verbs even further,
to the point where they now belong to
a syntactic category of their
The following question was also about do-support, but not specifically about your case: English questions and negation with *do* in syntax