How can I repair peeling finish on a faucet handle?
The peeling finish appears to be a metallic plating over a another metal or over plastic. 

You might be able to glue the lifted areas down using a cyanoacrylate type glue. You need to be very careful with these because they will glue your skin to a surface in a second or two. Such a repair would be iffy, and the handle may continue to peel in other areas. If there are small gaps, you could use a marking pen to mask them, but the coating is fairly transparent on metals (on plastics they tend to look more solid).

You could remove the handle, scrape all loose areas, sand with a very fine paper (or emery cloth), wipe with a solvent (such as mineral spirits) and then spray paint with an all surface enamel. There are a few plastics that resist even that type of paint (such as polyethelenes), but would not be likely used on a plumbing fixture.

Or you could try to get a replacement handle from the manufacturer.