A simple way to make dried up ballpoint pens write well again:
I always store them with the nib side down, sometimes even without the cap, so that the ink flows out of the nib a bit (and inside the penstand, but you never notice it if the penstand is black)
This clears the dried up ink from the nib, and makes the pen write like new.
No mess involved.
There are other options, like keeping the nib near a flame, or warming up the entire pen, but the method I have described does not involve any of the trouble.
Photo credit: amagill