Actually, wegen and zugunsten are quite dissimilar. The case of the phrase governed by wegen varies between genitive and dative (wegen des Wetters, wegen dem Wetter) depending on whether it is elaborated or colloquial speech, on the preferences and education of the speaker, etc.; additionally, the postposition can only govern genitive (des Wetters wegen). For zugunsten, however, it only depends on its usage as a preposition or a postposition. The preposition governs genitive, the postposition dative:
Sie haben zugunsten ihres Vaters darauf verzichtet.
Sie haben ihrem Vater zugunsten darauf verzichtet.
I would say that the postpositional use is less common.
For your specific question, this means that either mir zugunsten or zugunsten meiner are conceivable; zugunsten mir is ungrammatical. The genitive of personal pronouns sounds awkward, though, and is almost always avoided in present-day German. As already mentioned by Carsten Schulz and Toscho, one common choice is to undo the grammaticalization of zugunsten, which is a contraction of zu and Gunst, and say zu meinen Gunsten. In a similar way, zu unseren Gunsten, zu Stephanies Gunsten, zu wessen Gunsten? are possible. Outside of these fixed phrases, the plural of Gunst is no longer used.