I think what made Goldeneye so great was the gameplay variety and the combination of stealth and "run and gun." First of all, almost every mission was different (even if it took place in an area you'd already been through). Second, to survive harder difficulties, you had to be careful. When your cover was blown (e.g. a camera saw you or somebody triggered an alarm), you really had to worry. On top of that, harder difficulties weren't just about enemies that could take more damage. You also had to complete more objectives, so there was a lot of replay value (not to mention trying to unlock cheats and 007 Mode).
Also, at the time, there was simply nothing like the multiplayer available anywhere else. There were probably more than 100 different ways to play with friends, and the plethora of actually different weapons required a player to use a fair amount of strategy to be effective.
[EDIT] Then there was the whole "wait while two friends take each other down to low hp and then run in and kill them both." That was my favorite. [/EDIT]