This is the white one that looks like a USB flash drive (with USB connector), correct? It's getting old, so the most likely reasons are worn out USB connector or worn out battery.
When you connect it, does the light come on initially? Does the iPod still appear in iTunes? If it does, it's likely to be the battery. And if it appears in iTunes, you can try doing a Restore on it, from the shuffle's Settings screen.
If it does not connect in any way, it's likely to be the USB connector on the shuffle..
Oh, one more thing... To rule out the USB port on the computer as the cause for your problem, you should do the following, if you have not already.
Shut down (power off) your computer. Disconnect all USB devices (including hub), except for standard keyboard and mouse (if normally used). Do this to reset the shuffle
Start up computer, run iTunes, and connect shuffle to a direct USB port on computer.
Also, you can try connecting something else, preferably something similar like a USB flash drive, to the same USB port, to confirm it is working properly.