Repaint eyes of a stuffed animal
Probably the best thing is the acrylic paint uses for plastic models followed by a couple of coats of gloss acrylic varnish. This will be pretty durable and should give a good result. They paint system sold by Games Workshop is pretty good quality. 

You will want to thoroughly clean the surface before applying the paint alcohol is probably the best thing for this. It will also help of you scuff up the surface with wire wool although this does mean that you will probably need to paint the white parts as well. 

You may also want to mask of the fur around the eyes (assuming they aren't removable) you could do this with a piece of thin plastic cut from a drink bottle, food packaging or similar, Cut a large disk and cut a slit in it so you can slip it behind the eye.