PLayer one activates Puppet Master putting the "target player exhausts a character they control" effect on the chain.

Player two chooses Terra and Activates Terra to Place Terra's effect on the chain.

priority is passed and the chain begins to clear.

Terra stunns Puppet Master and KO's a location.

Puppet master's effect resolves....

Now this is the point of contention: Does the target player name the character to be exhausted when the effect is oput on the chain or when the chain resolves?

I'd go with when the chain resolves since the effect targets the player not a character.

For some reason I get dragged into these arguments at my venue.

The target player doesn't choose which character to exhaust until the effect resolves. The only "target" declared when the effect is put on the chain is which player you are targetting because that is all the ability asks for at that time. Secondly, remember that Puppet Master's ability states "exhaust a ready character he controls", meaning that if someone was to respond my activating Terra, she is not a legal target for the resolution of Puppet Master since she is no longer ready, and they must choose another character.