Anything turn-based, Civilization 5 or 6, Heroes of Might and Magic 3 or 5, Songs of Conquest for strategy. Pretty much any card games fall in turn-based category: Slay the Spire, Balatro, also all the pay to win hells of Hearthstone, yugioh, MTG:arena, they should all be very mouse-only on PC.

Old (think like Super Nintendo era) emulated RPGs could be mapped to only use a small subset of the keyboard for just one hand, worth trying out older Final Fantasies, Chrono Trigger, pretty much all older Pokemon games.

If their multi-button mouse has tons of buttons (also known as "MMO mouse"), I think realtime RPGs with mouse movement like Diablo II or Path of Exile could be playable as well, but this should definitely be tested with a trial cough pirated cough version before committing money to it. (Path of Exile is free though)

Visual novels if you consider those as games, not a big expert myself.

On smartphones: Shattered Pixel Dungeon IMO is fun and, in my experience, quite playable with one thumb in portrait mode

As for multiplayer aspects: Civilization 5 & 6 was pretty simple to set up a match with friends; trading card games obviously have multiplayer; Path of Exile is a single-player&MMO hybrid creature; Stellaris, Heroes of Might and Magic, Songs of Conquest, Diablo2(I think) technically have multiplayer but I have not tried it out (I've tried Heroes multiplayer locally, not over internet); Pokemon games have trading and multiplayer matches but you'll probably have to mess with emulator a bit to have it working.