Any disposable bedding (the kind you throw away each time) is going to be pretty expensive, because you have to keep buying it. Some options are cheaper than others, but if you have a large cage, you are going to use a LOT of bedding each time. There is no way I could afford to fill my 2x6 cage 2 inches deep with Carefresh each week, for example. It would cost hundreds of dollars a year.
Safe shavings will be cheaper, but you will need a LOT. I'm not sure how much it would cost to cover the floor of a 2x4 cage, or how often it would need to be changed. This information is on the forums somewhere, though - do a search.
Fleece with U-Haul blankets under it is one of the cheapest options (if not the cheapest). You will, however, need to spend about $50 upfront buying the fleece and blankets, and your mom will have to be OK with you washing them in the clothes washer (or prepared to drive you to a laundromat each week).
The cage will not cost you a TON of money - probably about $40 in materials.
The guinea pig itself will have an adoption fee (hopefully you are not planning to buy one from a pet store) of about $20-$30. Ideally, you need to adopt a pair. This will cost about $30-$50 from what I've seen on Petfinder.
Then you have food (about $10 per month for good quality hay and pellets, buying in bulk, plus groceries ($20-$40) per month, depending on where you live), and vet expenses (a basic checkup/visit is $40). I had over $300 in vet bills the first year I had my pigs, plus (very sadly) autopsy and cremation fees.
They really are not cheap pets, so please think VERY carefully about this.