A large part of this is from theory/reading .... so take from it what you will...
I'd perhaps start with what tyre or tyres you are looking at and they should have a suggested rim size or rim size range.
The wider rims will provide more support or stability to the tire and lessen how much it deforms under load / cornering making it feel sharper. Wider offset will increase the track which helps cornering but can also make your mini more of a handful (below).
The problem becomes this extra width often comes with an offset to the outside of the vehicle which is good if you want that stanced wide look but for handling it's generally a bad thing (but it does widen track).
The extra width means the wheel has more leverage over the steering so torque steer and irregularities in the road have a greater effect on the wheel you feel this like the car is trying to fight you.
Mini's were designed around 10" wheels and use the extra sidewall profile as part of the suspension, going to 12's isn't as bad as going to 13's but you are reducing the sidewall and therefore the amount that can be absorbed by the tyre, you MAY be able to get some of this back by using a rim at the narrower end of the suggested range.
Extra width and offset may need other flares so this enters into it as well.
The further out the wheels are offset the greater the leverage (as before) but this leverage also applies to all of the suspension components, by fitting 12's you're probably asking more of the suspension (ball joints, bearings, shocks, rack ends etc) (larger heavier wheel, less absorbsion though the sidewall, and generally heavier brakes) an offset would magnify this... that said they do work and survive. Extra weight and leverage may mean more is required from your shocks to contain it as well so they may need to be upgraded.
In summary if I was prioritizing handling I'd fit 10's if I had to go bigger I'd go 12's, next I'd try to get the smallest offset to the outside (offset in is better) of the car possible, then try to search for extra width after this, extra width will quite often come at the expense of this offset.
What is the car intended to do? is it a daily? weekend toy? or track car?
I remember a comment of Calver's saying bigger wheels accentuate everything bad about the mini suspension, and I promptly kept to 10's that said 12's aren't as bad as 13's.
Here's a link (there's I think another article in there that is useful):
http://www.calverst.com/technical-info/suspension-kpo-adverse-effects-of-fitting-wide-wheels/
