Well as cute as a mouse or rat may be, it would be in your best interest to remove it. We don't want you getting sick. If you really want to avoid killing it then I suggest the kind of trap that I used on squirrels.
It's like a long box that you put bait in one end and it has a trap door on the other so when the rodent goes in, the door shuts and traps it. Then all you have to do is take it somewhere away from your house and release it. No mess, no fuss, and you and the rat will both be happier.
The traps look like this.
http://www.northerntool.com/webapp/w..._988790_988790