Would it be possible to send money through the Lightning Network to an offline address?
This sounds like a web wallet, where the keys were managed for you and you simply interact through a website.  You'd have to trust the third party, but as we've seen on the main network, this is acceptable to a great number of people.

Just like with the existing Bitcoin network, you have to run things yourself to be trustless, but there are more convenient options if you are willing to trust (and pay fees to) other parties.