I guess the question title sums it up. Is there a time when it would be considered good design for an ABC to have data members? I have been wondering if there is a situation where this is OK. The only ones I can come up with all are static, and even then it's kind of a stretch.
I don't see why an ABC couldn't properly have per-instance (aka non-static) data members, as needed to support the methods it supplies to subclasses. Take the common case in which an ABC exists to supply a Template Method DP (the hook methods being abstract) -- if part of the function of the organizing method is to update some instance variables (for example, a count of how many times the method was called), then obviously those variables should also be supplied by the ABC. Can you explain better why you think that's bad design?!