Distance to a moving object
Measuring the distance to the mouse could be re-interpreted as locating the mouse within a 1m x 1m area. A 1cm accuracy could be achieved with a digital camera.

You don't say how frequently the distance needs to be measured, but a web cam could be used to get frequent updates. (A mouse isn't likely to move a huge distance in the frame-time of a video camera)

You don't say if the measurements need to be done in real-time, or could all be processed later off-line. So I'll assume real-time. It may be useful to both track in real-time and record the pictures for latter investigation.

Put an LED with a small hearing-aid battery on the back of the mouse. The LED should emit a distinctive colour. Shield the LED so that it doesn't shine brightly at the mouse. For example use a short piece of drinking straw painted black. A shorter wavelength IR LED, say around 850nm, would likely show up on a video camera. 

Here is a List of small battery sizes at wikipedia. Choose something which supplies 3V, and is as small as you can easily obtain so that the mouse can carry it. 

It will also need a small resistor to limit the current through the LED. This is a balance between keeping the current low, so the battery lasts a long time, and ensuring the IR LED is bright enough, so that it is easy to detect.

For example: this Osram SFH 4350 is 3mm in diameter. It has a typical forward voltage of 1.5V. Assuming a current of 2mA is enough, then from Ohms law the resistor would be (3-1.5)V/0.002A = 750 ohms. That happens to be a preferred value, i.e. you can buy exactly 750 ohms. I wouldn't worry about being too precise, anything from 680 to 1000 (1k) ohms should be fine.

As an alternative this Kingbright L-7104SF6BT is also 3mm. It's forward voltage drop is typically 1.35V. So the calculation of the resistor is (3-1.35)V/2mA = 825ohm. So the exact IR LED specification doesn't make much difference. A resistor between 680ohms and 1k ohms should be okay.

Use a video camera with a computer running some image processing software, for example OpenCV.

A Raspberry-Pi with its video camera add-on might be a way to do this if you don't have a spare computer to dedicate to the job. There is quite a lot of information about image processing experiments using a Raspberry-Pi on the web.