A worthwhile example is the problem of encrypted spam. In theory, spammers could use public keys to evade spam filters, and systems like PGP and S/MIME do not give you any ability to prevent that. On the other hand, an FHE system would allow Google to perform spam filtering on your encrypted email, and so that you receive both the email itself and an encrypted "spam or not spam" bit from Google. You can imagine this sort of thing be applied in other situations -- advertising, options modeling on EC2, etc.

Unfortunately, FHE is nowhere near practical enough to do that sort of thing. Maybe in a decade or two we will see FHE implementations used outside the research community.