Really? I guess it’s a Big-Pond issue. I [US] would at first
be confused by that answer: What don’t you mind? My
equivalent would probably be „I don’t mind which one,“
…and I thought it would be obvious to the Americans who are, after all, world-leaders in shortening words, names and phrases.
We just leave off the „which one“ because it’s self-explanatory.
although my actual answer [if I were in fact a coffee drinker]
would most likely be, „Either one’s fine.“
I would tend to go for that in another situation, maybe with people I don’t know so well. It sounds slightly more formal. With Mum „I don’t mind“ would be the standard choice. Or „Whatever you fancy“ / „Whatever you’re having“ / „Whatever’s easiest“, i.e. standard Brit I-don’t-want-to-cause-you-any-extra-work-type answers. 
Getting back to a point mentioned higher up the list, it is very important to differentiate between " I don’t mind" and " I don’t care". " I don’t care" is very much a snub and can sound extremely rude. Depends on the situation, of course, as ever. However, if my dear old Mum asked me what I wanted to drink and I answered with „I don’t care“ I would probably get my future told, even at my age. 
Ciao
MacD