You do not want to change the diameter. You can usually up the width a little without rubbing, but it varies greatly from car to car...and it will do nothing for ride quality.
What tires are on it right now? As in, brand and model? The hardness of the rubber and the stiffness of the sidewall can vary greatly between tires.
In general, a higher speed rating means a stiffer sidewall. S and T are often pretty similar. Most H tires start to get pretty stiff. Cheaper tires, often moreso; my thinking has always been that your cheap stuff takes a 'brute force' approach, usually resulting in tires that are overly-heavy and kind of 'under-engineered.' Whereas I know I've mounted Michelins with pretty tall sidewalls and H or higher ratings that was still pretty soft.
Short sidewalls tend to be unavoidably stiff. If you go to a big tire place like Discount Tire, they should have plenty of displays. You'll understand what I'm talking about if you feel the relative heft and stiffness of 'low profile' tires versus what's on your car. Also also the differences between tires of the same size...look at a summer performance tire with a fairly tall sidewall. Then look at their cheap S-rated off-brand/house-brand version of a similar size. Big difference. What costs money is balancing those two; having comfort without heavily compromising cornering ability.
BTW, for speed ratings, S < T < H < V < W < Z, IIRC.
The other thing is the tread pattern and compound. Look back at a generic Michelin 'touring' tire or something. Notice the assload of little grooves (sipes) in it. By comparison, more solid tread designs are usually louder and ride rougher, but the lack of 'give' increases responsiveness.
It's very hard to judge the rubber compound itself. Treadwear rating can be a good indicator, but like everything else I've said, it's just a rough guess. Most 700-800 tires might as well be coated with iron.
In summation...use intuition to pick out a good candidate for replacement tires, then read reviews. That's about all you can do. Don't mess with the size.