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I make a rescue attempt with a hero who is not protected from withdraw. My opponent blocks with Goliath. My original hero is withdrawn by Goliath's special ability, and I send choose Solomon as my new rescuer.

The battle is now a stalemate with Solomon (9/11) and Goliath (10/10).

For the purposes of initiative, who "played" the last card and why?

I'm hoping it's just that simple, that's how we played it 

wouldn't goliaths ability only complete after Solomon has finished his ability due to goliaths ability allowing for Solomon to be added to battle?

Yes, but 'playing' a card happens when the ability is activated, not when it is completed. So even though Goliath's ability isn't complete until after Solomon's ability is activated, Solomon was still the last card played.