It sounds like your mom's home isn't the greatest environment to leave your daughter in, and I also think it might be too hard on you emotionally to leave her. Plus, you'd have to go home every weekend to see her, and it would cut into studying and/or socializing time.
If you are going to a school where students don't usually have children, you might find your daughter gets "adopted" by all of your friends, and they may be able to help you out in a pinch. Also, since you're saving money by going away, try to find a list of people that could babysit-a great source is child development programs, teaching programs, or nursing programs-many of them have flexible scheduling as well.
Finally, don't think you have to go to med school right away-it will be there; your daughter won't be young forever (I stayed at home when my kids were little, and started med school when my youngest was 4). It is much much easier when they are older! If it is too hard, consider taking a year off from college; work part time doing research or something medically relevant.
Good luck-and good for you for thinking so much about your daughter's welfare.