1. The Legend of Zelda: Ocarina of Time
Although The Legend of Zelda: Majora's Mask looked better than Ocarina of Time, there is something about Ocarina of Time that will live on in our hearts forever. The Deku Tree, Jabu-Jabu, and the Water Temple. The graphics were great for time and it was one of the best games of 1998, in fact it was voted The Best Game of 1998 if my calculations are correct.
2. Super Mario 64
This was probably the first game many people played on their Nintendo 64 system. They took the fun of roaming around from Super Mario RPG: The Legend of the Seven Stars and brought back the original storyline from the original Mario games. Save the Princess from Bowser. This was also when Princess Toadstool's name was changed to Princess Peach.
3. Conker's Bad Fur Day
From Rare came probably the most crude game of all time. A definite third-party game that you wouldn't expect on a Nintendo console. A remake was made for it on the XBOX (or the 360 I can't remember) but people still try and track it down on the old N64. If you pass by it I would suggest picking it up.
4. Mario Kart 64 and Super Smash Bros.
Although it's the number 4 spot, both of these games deserve equal treatment. For they were the best games to play as multiplayer games. Mario Kart 64 came after Super Mario Kart for the Super Nintendo and Super Smash Bros. was brand new. Both of these games were usually played whenever friends came over to visit and still are today. When Mario Kart: Double Dash and Super Smash Bros. Melee were both released on the Nintendo Gamecube multiplayer status among those games continued to thrive. It is no surprise that Mario Kart and Super Smash Bros. seem to be in everyone's Nintendo 64 collection as multiplayer games.
Even though Banjo-Tooie was a nice game and had a lot of new features at the time, but Banjo-Kazooie, like Ocarina of Time, will always have a special place in our hearts. Also Banjo-Kazooie seemed to have a bit more comedy than Banjo-Tooie, while Banjo-Tooie had a good bit of laughter in it, it seemed a bit more serious.