I'd hope they'd fix the basic movement controls and camera. If we're going to be spending a lot of time walking around sometimes cramped environments and trying to position ourselves to focus on very small items scattered on the ground in the game world, movement has to be precise and comfortable.
The cover system and combat is classic Rockstar, which is to say its clunky and still way behind the times. If the game is going to throw in combat and racing as the fun and exciting bits, those aspects of the game HAVE to be nailed, and, well, fun and exciting.
In terms of the cases themselves, I'd hope they'd give more options in how to approach interrogations, remove the fail conditions (allow retries but allow the possibility of continuing the story even with botched cases), allow us to control our level of aggression in questioning, outside of whether we believe the subject or not, and provide more game elements to the questioning besides purely determining truth or falsehood.
Then give us a change of pace between cases. Don't just stack them up on top of each other in sequence. Put them as spots on the map, so that when we arrive in a certain area of town they unfold, and/or provide us with some alternate gameplay (not cinematics) for us to engage in after finishing a case. Maybe we meet up with other detectives at the bar. Maybe we go home, something, anything that gives us insight into the world and characters, but that is playable.
Last bit - give us a pathway through the game where we don't have to kill except under extreme circumstances. Let us take down offenders by non-lethal means or allow us to take limb shots. Reward us for bringing suspects in alive.
With that kind of effort I'd be all over a sequel.