Now for most of us, we just toss out the dull seam ripper and purchase a new one. I never considered this as my rippers only cost a dollar or two. Maybe someone has a solution for you other than purchasing a new one. Budster of little help.

Basically, you can't. Toss it and buy a new one. They're cheap.

Ditto what the others have said~~~~~~~~~~~~~~Buy a new one.
Better yet when joanns has their 1/2 off notions,buy a couple of them!!
Kathi

Since I recently learned how to sharpen knives and scissors, I was curious about this... I think you'd need a very small (1/4") round file (which would probably cost more than a new seam ripper). Notice the cutting edge is beveled on one side, so you'd want to gently run the file in one direction on that beveled side only.