What do you call a word like controversy, which has more than one pronunciation? I don’t think there is a widely accepted term, so I would like to suggest polymorph. The word is already used in various disciplines, such as zoology and chemistry, to mean an entity that appears in two or more different forms. As far as I know, it has not been used in linguistics.
My thinking about such words was triggered by hearing the word byzantine used in a television programme. The OED gives the following as one of the meanings of the word:
Reminiscent of the manner, style, or spirit of Byzantine politics; intricate, complicated; inflexible, rigid, unyielding.
and gives only two pronunciations for GB: bɪˈzæntaɪn and ˈbɪzəntaɪn. However, the first vowel can also be aɪ and the final vowel can be also be iː, so it’s clear that the word is a pretty good example of a polymorph.
An even better example is the word transitionally. I once calculated that this has over 70 different pronunciations in GB, but Jack Windsor Lewis has since come up with 100+ (if memory serves me well). I don’t know if JWL has reported on this tour de force in his blog. Perhaps you could let us know, Jack?