I've been really excited about the potential use of HE for private messaging. Today the most anyone does is end-to-end encryption, which does a great job protecting what you're saying, but it fails to protect who you are talking with. HE has the potential to change that.
The Wikipedia demo[1] starts with a big download because you're fetching an index of articles. Subsequent requests have you send an encrypted one-hot[2] vector making the article you want to read. The server does an encrypted dot-product of the vector with the vector of articles, returning just the encrypted article you're looking for.
A messaging system could do the same thing, where your vector selects member identifiers or public keys.