Your mother would be able to obtain a FHA mortgage loan with her score. Her interest rate would be a bit higher than the 3.5% most would be qualified for with a better score than she.
She should contact and apply for a mortgage loan through a mortgage lender that is authorized to do FHA mortgage loans.
Mortgage underwriters base their approval on a ratio. This ratio is obtained by the amount of annual earnings of your mother and the amount of debt listed on her credit report. The brand new car would be included since this would be listed on her credit report.
Her ratio should not exceed 35% of her annual income. This ratio would determine the amount of house she would be qualified to purchase, whether it is a $250,000 or $100,000.
I am always reminded of a statement when someone mention having a co-signer or co-owner. Unless these two people are married, very seldom do these arrangements work out. The saying is that oil and water never mix as friends, relatives and finances are like oil and water they don't mix very well either.
Your mother should contact a mortgage lender that does FHA mortgage loans to be pre-approved for a mortgage loan. This pre-approval would indicate the amount of house she would be able to purchase.
Once pre-approved she would then be able to contact a real estate agent to begin the search for a house.
I hope this has been of some benefit to you, good luck.