Hi there. Try Elephant, Diamond, Admiral (all the same company) they are good for young drivers, both of my girls started with them. I think Elephant do a 10 month policy that gives you a years bonus after just 10 months.
Add a parent or two on you policy, as long as they have a good driving history, that should make a difference.
Can you park your car off the road at night? That usually makes a bit of difference.
Restrict your mileage, the less miles you do the lower your premiums will be. Some companies also offer a discount for not driving at certain times.
Go for a bigger excess, that will reduce your premiums but bear in mind that if you do have an accident then you will have more to pay.
Check out the extras that get added on, do you need a courtesy car if you had an accident, you you need legal cover? All of these things, whilst are handy can push the premiums up.
You also need to bear in mind that if you get your car and insure it before you pass your test then your premiums will go up when you have passed as you will then be driving unsupervised.
The other thing is, a MINI One will be cheaper to insure than a Cooper if you are not bothered about a Cooper.
Good luck and post some pictures when you get sorted.