Hi there! I've never heard "beg off" or "cry off", in this context or any other. So the first two do not make sense to me. The third one is closer, but I would probably phrase it as "My cousin is thinking about backing out of going to the wedding" (although simply saying "My cousin is thinking about not going to the wedding" would be simpler). Unless he / she is the groom / bride, then I'd say "My cousin is thinking about backing out of the wedding".

Cry off is not used in American English. "My cousin wants to beg off going to the wedding" sounds better. "Beg off" usually takes a gerund: I need to beg off going someplace, but for some events it's omitted like "I need to beg off lunch today". "Backing out" implies that the person has made a significant commitment: 'I need to back out of this deal'; 'I'm backing out of this production'; 'The bride backed out of the wedding'.

Hey, beg/cry off seem to be good solutions but phrased differently: My cousin wants to beg off (going) the wedding. My cousin is invited to the wedding but he/she wants to cry off.

Thanks, Nancy.

In my opinion the 3 sentences mean the same ,that the his or her cousin is thinking ask to be excused to not go to the wedding.