If just the outer jacket is cut, and there is no damage to the shield (the metal braid or foil that surround the inner wires) then just wrap with good electical tape or liquid electical "tape" and go.
If there is real damage, I would suggest nothing short of replacing the cable with a new USB cable (buy one and cut one end off). It might not be easy to attach to the mouse, but no doubt it can be done if you or someone you know knows what they're doing.
Patching the cable is probably a bad idea if any of the wires or shielding is damaged at all.
Important Warning: USB carries power. If you screw it up, you can damage your motherboard or who knows what. Not for the unhandy.
If you must, then:
Cut the outer jacket of the cable lengthwise for about 3 inches (carefully; don't damage the braid or foil);
Repair the broken wire inside the shield; if just the outer jacket is exposed, I would just use a blob of an appropriate goop to protect exposed wires from short circuits;
Wrap the repaired area with foil which completely encloses the repair and makes good contact with the braided shield;
Tape over the slitted area you made to get inside the cable jacket in the first place.
USB is not particularly robust when it comes to poor connections or bad shielding; the USB 1.1 spec doesn't even allow cable lengths beyond (I forget the exact figure) about 5 feet. Thus those annoyingly short keyboard cables.
Since your whole USB bus will be affected by poor cabling, it's no place to try 2nd rate repairs. You will have problems if it's not just so.
If after all that, you want to try a repair go ahead. If not, a new mouse is really the way to go.
thank you. the shielding is cut through. i think the smartest thing is to replace the entire cable. i could probably do it myself depending on how the cable connects on the inside of the mouse. ill see if i can even open the mouse without breaking anything.
For those in search of this ever-popular question. I got my answer from eHow.com. The main point is that there is one hot wire (usually red), one ground (usually black) and the rest two (usually green and white) are data cables, called D+ and D-.
My little kid (4 y.o.) cut the mouse cable as an intelligent way to prevent his little sister (1) from "disturbing" him (compare that to my gentle way of using software to lock the mouse or creating an account for him with limited powers and logging in with his account!)
After spending many frustrating hours with a local technician, I turned to Internet and landed on eHow.com, where I found the solution. What's more, I found upon closer look at the mouse that the nice people had actually marked the points with V, G, D- and D+!
So I just connected the red wire to V, black one to G and then tried the green & white to get the correct match with D- and D+ and guess what? It worked!
I'm glad that my wicked 4 y.o. did that. Just if you allow me, I'd recommend thanking God on every good and bad thing that happens to you (according to the theory that God is absolute good) -- you will find peace of mind, at least. Now I thank God again that I thanked Him when my expensive mouse was cut by my own pliers by my own loving son! Thank you.