Sounds like you don't have the pigs anymore. I can understand if you haven't looked after them well in the past why your mum would be apprehensive getting you a bunny. Buns can live past 10 years, so it's good of your mum to be so caring.
I think though if you know in your heart you are going to do everything youc can to give the bunny a good life and do your research and are prepared to clean the hutch out daily- if you lino it first it will make cleaning easier.
It is best to have a minimum of a 6x2 hutch with a 6x4 run attached. It is also best to have two buns rather than one, unless you are going to be there all the time, buns get lonely and need company of there own kind.
Good luck in persuading your mum
& hopefully you will get some good advice on the forum.
I would say also make sure they are both insured whether you get two or one. So many things can go wrong with buns. They also need six monthly vacs and annual ones for mixi and vhd which is £45 a year usually with vets per rabbit.
The hutch and run setup may cost you no less than £200.00 if you buy something of the right size. Round about £100 for the hutch and same for the run. Other people may be able to give better suggestions- i.e if your mum would let you have a playhouse and run etc.
Hope this helps x