There are three similar expressions:
- gyda'n gilydd - (we) together
- gyda'ch gilydd - (you) together
- gyda'i gilydd - (they) together
The original base word in the expressions was cilydd, meaning 'a fellow, a companion', although it is not used with that meaning nowadays. The version gilydd with its fixed mutation is what remains in those three expressions above.
Note that the expected * gyda'u gilydd does not exist, although it is a common error. gyda'i gilydd is the correct version for '(they) together'.