I am in joint tenancy with my mother, the co-borrower on the house. She has never lived here, nor has she paid any of the mortgage for the entire 7 years.
She wants to now change from joint tenancy and get her name off the loan. I do not want to sell the house, and cannot assume the loan because I am not approved by the bank due to credit. I have made every payment on time for 7 years and do not wish to move myself and family.
What options do I have to keep the house?
You both are stuck, really, if you cannot come to an agreement. The best option is to work out a compromise.
I understand why she would want out of the deal, as it is a bad situation for her. She is taking on all the risk of owning a property, but with no benefits.
I would suggest you offer to pay her some amount of rent in addition to the mortgage. This may make her reconsider. You have essentially been using her credit for free, as you could not get approved for this loan.
Expect your relationship to deteriorate if you do not work this out. You might have to accept that you need to buy a house you can afford.
Are you sure you cannot assume the load yourself? Talk with your bank, or see if you can re-mortgage with an other lender. If you have kept up with mortgage payments for 7 years, that should make you a good loan candidate unless you have a lot of bad credit from other things.
Otherwise you will just have to talk to your mother. She probably does not want to force you to sell the house, so she may be willing to stay on the loan if you are seriously working towards getting her off.
