When only half the string lights up, it's usually because of a single light in the bad section. I've had this happen to me a few times now. I've fixed the problem by individually pulling each light and inserting a light you know is working. You're done with this exercise when you find the bad light and the string comes back to life. If the bad light is the first one you replace, it's an easy fix. If it's last one in the string, you'll be wondering if you shouldn't have just gone out and bought a new string.
The toughest thing about doing this is getting the light covers off to expose the LED. On most of my strings the covers are hard to pop off. I end up with a raw thumb if I have to pop several to find the bad light. One way to avoid this is to get lights with covers that screw off.