Banjo and Kazooie are really one of the last 3rd party characters that would really bring the hype that’s Sonic and Megaman did.
I came into this thread to make this exact comment.
I feel like there's a decent chance we just don't get any new 3rd parties in Smash 5. There's not many characters left that are THAT important to Nintendo's history (Sonic, Mega Man) or gaming history (Pac-Man, Ryu, Cloud) as a whole. Additionally, there's not many third party characters that people would hype over in large amounts. Most of the third party characters people bring up are a stretch for one or both of those categories.
Simon Belmont? Yeah, Castlevania was big on NES, but he hasn't been on Nintendo lately and not THAT many people want him.
Shovel Knight? Yeah, he got an amiibo and is really freaking popular, and could represent Nintendo's new focus on indies, but he's on every platform now. He's not close with Nintendo specifically.
Banjo-Kazooie is very widely known, especially thanks to YouTube reviewers like JonTron helping popularize it. The game defined Nintendo's Rare era of the 90s and the 3D paltformer era of the N64 with everyone trying to make the next Mario 64. Not to mention Odyssey clearly took some inspiration from Banjo-Kazooie, probably just as much as from Mario 64.
I personally believe that if Banjo-Kazooie wasn't owned by one of Nintendo's two direct competitors, they may very well be this Smash's Sonic or Mega Man, with everyone hyping them up as the 3rd party character that NEEDS to be in.
TL,DR: Banjo and Kazooie being in Smash would be bigger than the majority of the other third party suggestions people have for the next Smash, and would rally more support if they weren't owned by a direct rival in the console wars, which people think makes them near impossible.