How can I show rental history for someone living in my home?
How can I show rental history for someone living in my home who is not on a lease but pays part or all of the mortgage? Whatever the answer, will it be suffice for the next rental agency or mortgage company when she tries to buy her own home or rent an apartment/housel?

If the person is living under the same roof as you, they’re typically considered a roommate, not a renter, but this is a bit of a gray area amongst mortgage lenders. For the purposes of procuring a mortgage for her, many non-prime lenders don’t mind, and will simply require that you provide 12 months’ worth of canceled checks to prove her rental history with you.
With regards to a new lessor, her residence in your home could count as rental history, but would depend on the property management company to which she applies.