It's a perfectly good place for this topic.
I did half-lease in past. When you'll talk to the owner (mom) ask what if horse will be unridable for some time (things happen: colic, lame, allergy, anything). Will you get your money back? Will they be extended for the next month? That was my big issue when I half-leased: horse got a bad allergy somewhere and wasn't ridable. However I never got my money back.
Also be sure if you get tack coming with the horse (so you shouldn't worry about the fit etc.).