Spending my 7 free days of WoW - what are some fun quests at level 77?

At 77, you don't have a whole lot of options. You could go back and run through some of the lower level areas (I personally found Thousand Needles to be quite a bit more interesting, but I am on the Alliance side). Or, you could quest out in the Zul'Drak/Storm Peaks/Icecrown areas.

It would be in your best interest, however, to avoid the new Cataclysm content, because the mobs it quite a bit harder, to the point where some people going into those areas at level 80 found it somewhat difficult.

Roll a new toon. Some of the new starter zones are great. I assume you can't do Worgan or Goblin but are well worth a play through if you get the chance. All the old world races have much improved starter quests. They really give you an introduction to the world story, vehicle mechanics, progressive story telling and quest that don't send you through the same spot 5 times. Undead is especially epic. You go from unbirth to discovering a covert attack to full blown invasion and war in just 2 zones. Lots of nasty stuff going on. Dwarfs have very strong first zone with a civil war/dark iron invasion, but after that I switched to the human lands which feel very well done.