Could start out that you're a poor mechanic in debt to a big crime boss (or whose sister is being held captive by a big crime boss) and you're waiting for his call--he's sending you on a big job to wipe out a rival gang boss whose headquarters are in another town. You have different ways to get there -- each one poses different problems, like you need a truck to blast your way through a barrier. You're a good kid so you don't mind taking out another bad guy. But you're set up. The crime boss is using you to transport something illegal and you're going to get killed when you approach the rival boss--you're really just making a delivery and you have to figure out how to outwit both sides. F.E., you're delivering an illegal shipment of explosives, guns and ammunition. All that could come in handy when fighting off whatever. But small explosives could accidentally explode causing damage to your car (or you) and then you have to stop for repairs. Or they could fall out and explode, taking off a little bit of wheel, etc. so you have to stop to repair your car.
How many cities involved? Shortcut could be through sewers. I love sewers! Through tunnels that connect the two cities because they're separated by a river---you could die if the tunnel is blown up and water comes rushing in.
You could decide whether to turn both crime bosses over to the police. Or somehow sell one out against the other, depending on which choice you make.
That's just a thought. That's GTA without being too evil. : ) You could have a girl friend/boy friend who gives you tip offs over the car radio. Just throwing stuff out. What comes over the car radio could be a white square up top left with a little bit of text that changes as things happen.
If you're faithful to the original, there would be little side missions, like helping someone steal a car, respraying your own car to avoid being caught by the police when you run over someone in your main mission or accidentally kill someone. Maybe you could wipe out some of your crimes from a corrupt vice squad guy who would "pardon" you for a certain amount of gold.
You could have a policeman's face up top to represent number of cops chasing you and the number next to it goes up each time you cause property damage or accidental bodily injury.
How complicated do you want to be?
I'd rather run over chickens than people but I realize this is GTA.