How would I go about buying half interest in my mother's home with her carrying the mortgage? What is the least amount of interest she can charge?

Finally who could arrange this for me?

You would first have to work with her mortgage lender to agree to the sale.

If they agree, she can give you a quit claim deed from her to you and her as joint tenants (with right of survivorship). You give her a promissory note detailing what you will pay.

You can check the IRS website for guidelines on what interest can be charged by family members.