Can I deduct a vehicle that my mother cosigned on, but I make the monthly payments? For self employment.

You can deduct those expenses if you are also liable for the loan. 

Is your name on the title?

You don't actually say who the owner is (whose name is on the title).

For the standard mileage rate, you must be the owner, because the standard mileage rate includes depreciation and only the owner can deduct depreciation.  If you are the owner, it doesn't matter if you have a co-signer.  That's just someone who promised to pay the loan if you don't. 

If you aren't the owner, you can use the actual expense method.  For the actual expense method, you can deduct your actual expenses, but this requires keeping track of all the vehicle mileage (personal as well as business) and all the vehicle expenses that you actually pay for (repairs, maintenance, fuel, insurance, etc.).  You would not include depreciation if you aren't the owner.  The actual expense method is described in chapter 4 here.

https://www.irs.gov/pub/irs-pdf/p463.pdf