The target is chosen at random as part of putting the ability on the stack. Players can respond to the ability knowing what the target of the ability is.

If Scab-Clan Giant leaves the battlefield before its ability resolves, no fight will happen and neither creature will deal damage. Also, if the target creature is an illegal target when the ability tries to resolve, it won't resolve and no fight will happen.

To choose a target at random, all legal targets must have an equal chance of being chosen. There are many ways to do this, including assigning each possible legal target a number and rolling a die.

In a multiplayer game, you don't choose a single opponent at random. Any creature controlled by any of your opponents may be chosen at random.