I think you are mixing and matching "family" and "type" a bit loosely. It's fundamental to Revit's whole category/family/type paradigm that within a category (say, furniture), family provides the basic geometry and fixed dimensions of a related set of objects (level seat, angled back, curved arms, four 2" x 2" legs) , and that type provides the dimensions that vary: L, B, H.
Families have nothing to do with variable dimensions, types have everything to do with them. So, within Revit's paradigm, what you really are asking for is that the name of the type (not the family) changes each time you change one of the L,B,H dimensions. The family name remains the same (Outdoor bench with angled back and curved arms) for all of its dimensionally different types (60 x 24 x 36, 72 x 26 x 42, etc.)
That said, this would indeed save a lot of time and frustration. I often go into "Edit Type > Duplicate", and change the dimensions but forget to change the name, or vice versa. Your request would get tricky to implement, however, if a dimensional parameter that is not included in the name of the type is changed, like seat angle, or glass thickness, or what have you. That type would be dimensionally different, but still have the same name (well, with a "2" after it) as the one it was duplicated from.