Threw together some ideas for their kit. Tried to make stuff that was distinctive and unique to them, no idea if it's overpowered, underpowered, etc etc. Optimally I think every character should be a little bit OP in some ways and a little bit UP in other ways.
Defensive Abilities
I've given B-K two tools that make the way they maneuver and approach different from other characters.
Wonder Wing. Instead of shielding Kazooie shields Banjo with her wings. Cannot roll as a result however is invulnerable, deflects items and does damage on impact. Shield lasts as long as a normal shield would however recharges much slower. Can jump while WWing.
Feathery Flap. Instead of air dodging Kazooie does a feathery flap. Basically an air dash only with it allows you to hover in place at the end of it.
Down Attacks
In the games a lot of B-K's kit is used through ducking. In Smash only a few down attacks exist but I've decided to unify them along a theme.
Down Tilt: Blue Eggs. Spamable ranged attack.
Down Smash: Grenade Eggs: Powerful ranged attack.
Down Special: Ice egg: Ranged attack that applies a slow. (Fire and Electric as customs)
Down Air: Beak Buster: Sends you flying downward. Big damage, spikes, if you miss you don't stop. Maybe you can Feathery Flap to cancel it so that you can play mindgames with that. Are you going for it or are you going to go for it after a FF?
Forward Attacks
The forward stuff would be more banjo spare the smash and draws more influence from basic moves than ducks.
Neutral: Claw Swipe. Not very good damage but you and your target move forward as you use it pretty quickly.
Forward Tilt: Forward Roll. Good for slipping behind enemies while doing damage to them at the same time.
Forward Smash: Beak Barge. Slides forward big distance while doing. Like Bowser's Forward Smash.
Other Details
Rest of their kit would back up that stuff. Banjo and Kazooie would be about delivering good damage at just the right time and being very mobile the rest of the time. Good at getting into position and good at closing. Probably also have some more stuff that you can spam to work up damage. Dunno what recovery, there'd be a choice between a very recovery focused flight or a very aggressive beak bomb. Maybe one would be stock, the other custom. Dunno. No idea if this makes any sense competitively, don't follow that scene.
By the way there's a hashtag for this now. Only me, DKVine and a few others have used it, maybe you guys should as well. Can we get Banjo-Kazooie into Smash? #YesWeKazooie