thelounge is an web based IRC client with persistence like irccloud, except that thelounge is open source and self-hosted.
A web-based client is not going to provide adequate persistence on a smartphone.
Lounge works fine on a smartphone because the actual IRC connection would be handled by a server where you host it, not your own phone.
Do you get notifications when you are mentioned on your phone though?
So can this be solved for free? Is it even a good idea to solve it?