Martha has never considered herself particularly spontaneous; she’s the eldest daughter, the problem solver, the mediator. Sometimes she feels more like the parent, trying to sort everyone out and keep the peace.
Maybe that’s why, when the Doctor asks her to go on a trip with him, she hardly hesitates at all. Doesn’t she deserve a break from being responsible and reliable? Doesn’t she deserve a bit of time to do what she wants instead of spending all her time juggling her medical studies and her family?
Besides, he can travel in time; she’ll be back before anyone misses her.