She might really think you're going to be inconvenienced and truly doesn't want to impose upon you. Perhaps if she hears from her son that Speedy won't be an imposition she might feel more comfortable.

My sister-in-law might move here for a year while her husband is in Egypt... it has taken me a few weeks to convince her that she CAN bring both her dog and cat and not worry about imposing. She thinks just having her here will bother us. Why should it? After constant "noooo problem" and "we insist on it" she is finally convinced we mean it when we say that the dog and cat are no problem (she wanted to give the cat to a friend, the dog would come along.)

Keep up trying to convince her and she'll eventually agree.