I run ZNC on a $5/month cloud server. It lets me use fancy GUI clients like Textual 5.[1] ZNC replays everything that happened while my laptop or phone was sleeping. Also, I can connect multiple clients simultaneously. Lastly, znc-push[2], alerts my phone when people mention or PM me.
I'm not a big fan of terminal-based IRC clients. Most text on IRC is prose made by humans, meant to be read by humans. Terminals use fixed-width fonts. That's handy for structured stuff, but for prose is less information-dense and less readable.
There is also some GUI clients that are designed with the server/client model. Essentially you have a little daemon running that is connected to IRC constantly and then you got a GUI you hook up to it. Personally I use Quassel, but there is also Smuxi (which I couldn't get working for some reason..).
Have you seen QuasselDroid? It's the Android client for Quassel, I'm working on a Material redesign for it right now ^_^
You should also try the quassel webproxy (a web interface like IRCCloud) and the quassel-search tool, which allows you to search through the whole logs of everything.
Or you can just use quassel, which has clients for android, iOS, windows, Mac Linux and the web, works exactly like IRCCloud, is fully open source, and is able to run on a 2$/month webserver.
That cloud server does other stuff, but the cost isn't really a factor for me. IRCCloud doesn't let me connect with an IRC client. I can only use the mobile apps or my browser. Many don't mind that limitation, but it's a deal-breaker for me.
