If you have a public DNS entry then you can get a certificate from Let's Encrypt using the DNS verification method, even if the name points to a private IP address. DANE would also be an option if it (and DNSSEC) were more widely implemented.

Local name resolution (mDNS) could take a page from what Tor already does and encode the public key fingerprint into the domain name. If the key uniquely matches the domain (perhaps just the first part, e.g. b32-key-hash.friendly-name.local) then you automatically get the equivalent of domain validation. While, by itself, this doesn't prove that you're connected to the right domain, by bookmarking the page and visiting it only through that bookmark you would get the equivalent of trust-on-first-use. Browsers would just need to recognize this form of domain name and validate the key against the embedded hash instead of an external CA.