I have a high end mouse and it'd be silly throwing it away or even sending it away for a chance to be fixed in weeks when one could just replace the cord with at most some soldering.
I'm not aware of such products. However, mice run on a low clock of 1.5 MHz instead of 12MHz typical for USB devices. Also the power lines in mouse cable doesn't need to carry 500 mA specified by USB, only about 30mA. This means the requirements on mouse cable aren't nearly as stringent as on general purpose USB cables, allowing to make them cheaper, thinner and more flexible.
If you look inside a normal USB cable, you will notice that it's two twisted pairs, and runs inside a shield. If you look inside a mouse cable, they are usually straight cables, and thinner, and unshielded.
The thicker part is for relief. It's supposed to be slightly more rigid than the cable and has the intention that the cable doesn't bend too sharply at the spot where it enters the mouse, prolonging the time till the cable breaks. It's necessary for thin cables, but is not necessary for thicker, less flexible cables.
Safe choice number one, is identify the point where cable broke, and shorten it. Chances are, it broke right where it enters the mouse or the relief. You may not even need soldering skills because the plug inside the mouse is usually crimped, not soldered.
Safe choice number two, take an inexpensive mouse and use the cable from that.
Unsafe choice number three, something i haven't tried yet, use just about any other kind of cable, like from a headset. Care needs to be taken to use such cables which prove solid in long time use. Often, headset cables are made not with solid wire insulation but by lacquering the wire strands. The lacquer will rub off eventually and there will be a short, which can damage the computer - whether there will be any damage will depend, besides luck, mostly on the quality of the PSU but also on the mainboard. Often, neither low-end nor gamer (fused rail) PSUs provide sufficient protection, but there are PSUs designed for safety which do.
And don't forget to complain. Making a cable which will not break in your lifetime isn't rocket science, and that's something all mice manufacturers should do, for any mouse that retails for 10$ or more.
I'm unaware that there are replacement cords, but you can always buy for example USB A-B cable (or A-whatever), cut off B side and use the A side as a replacement cable. The greatest difference between USB cables is if cable has shield inside or not. There are several sizes for wires inside of the USB cable, but I don't think that the difference would be a problem. It is also difficult to determine the exact size of wires when purchasing the cable, but if you're uncertain, go for thicker cable.
I replaced USB cables on several joysticks and mice and haven't had any major problems. I've heard that (but I've never seen any) some cables do have non-standard color schemes, so it would be best to confirm connections with a multimeter on continuity testing if you have one. Just cut the existing cable in half (but confirm first that it's the cable!) and see which pins on the connector go to which wires.
Also, be careful with ferrite beads which you may find inside the cable. It is important to have exact number of turns of wire going through them, so that they work properly. Also, make sure that you pay attention to problems which can be caused by cable tension. It's a good idea to somehow secure the cable on the inside of the device so when the cable is pulled, the solder joints aren't stressed. I've seen some people tie the cable into a knot inside the device's case, so that the knot is pulled. I don't think it will have major impact on the signal strength, if the cable isn't very long.
Here's my question on electronics.se about a similar problem. There, I was trying to repair existing USB cable with damaged connector, but some advice I was given could help here too.