The first Stanley Cup I remember staying up to watch in it's entirety - the Oilers putting an end to the Islanders 4-Cup dynasty in 1984. I was 8 years old at the time but I remember clearly the look on Messier's face when he got handed the Conn Smythe Trophy and Gretzky almost falling while putting the Cup over his head.
There's a whole slew of full historical NHL games on YouTube now and here's one of the best - in it's entirety from the original CBC Hockey Night In Canada broadcast - Game 5 of the 1984 Stanley Cup Final. I still get chills watching this today. The great Bob Cole calls the game, as he'll do tonight for Ottawa's first game against Montreal in the 2013 playoffs.