Some of the things I'd like to see with StarFinder are...
Fractional BAB system. This would be a bump to multiclassing which seems more fitting in a space faring adventure where everyone needs to know something about everything.
Core Rules based equipment modification system. I'd like to see a simple system on gear modification akin to the one used in Star War SAGA. Customizing equipment helps players get attached to gear and could be a good substitute for magical items which I'd like to see more as artifacts then common.
More standard attacks then full attacks. I'd like to see feats encourage moving around to find cover and taking one shot rather then multiples in a round. This means feats like Rapid Shot should be changed to be more Vital Strike like and that Cover bonuses to AC and abilities granting dodge bonuses should be more prominent then armor bonuses.
Mecha.
Heavy Weapons. One of my favorite things in Star Wars stuff is a full armored mando with a repeating blaster, I'd like that to be a viable build. Not broken, but viable.
Grenades. I'd like to see a feat chain that allows people to make grenades bounce before exploding after they hit the ground. Or let's it roll to get a few more squares. Or even feats that allow more control on the random direction die.
Mecha.
A class that is kinda shadowrun rigger-like with the ability to control turrets or robotic companions.
A blaster class that more 3.5 D&D warlock with a techno vibe.
Re-Done Chase Scene Mechanics that are more open ended and group based rather then individual. I'd like to see players who blow up the door when they go through not have to see the players after them still have to beat the door DC.
An Escape From... scene system. Like a chase scene but to avoid getting caught in the exploding ship, or the black whole, or that skeevy dude who hangs out in front StarHortens coffee...
A bartering system that is more then just opposed diplomacy rolls. For example diplomacy could get you a 5% discount, but using diplomacy with a blackmail info (knowledge: local) plus bringing your street tough with you (intimidate) could be more. Of course failing that blackmail check could make things more interesting, and should.
Flumphs. Techno Flumphs.
Mecha.
I want Starship Trooper (book) type suits of armor!
Space Obstacle course scenes. Who doesn't like having to dodge asteroids!
Pan Galactic Gargle Blasters and Gin and Tonics.
A lucky base system that allows for crazy luck opportunities to avoid death, or achieve great success. Maybe a permanent sore that depletes as you use it and can only be regain when accomplishing amazing feats without luck?
That's my spitballing for now, if I think of more I'll post again.