Perhaps older RPGs or newer RPGs that feature turn-based combat.
Trails in the Sky:
https://www.youtube.com/watch?v=kWQfycg4uZ8
Trails in the Sky SC
https://www.youtube.com/watch?v=73nBm_qRNEo
Pier Solar
https://www.youtube.com/watch?v=sz-2rXF7KjU

Mini Metro. You will be amazed at how much time in 5-15 minute chunks that game can eat up.

Day of the Tentacle. Actually, if you like point and click and story games any of the old Lucas Arts games like Sam and Max, Full Throttle, Monkey Island, Grim Fandango are games you most likely will enjoy, at the very least, they are worth taking a look at.

I know you said not the most common strategy games, but Roller Coaster Tycoon Series and Sim City are both pretty great, just in case you forgot or didn't know :)

In all honesty, how is that game? Is it fun?

Also, is that game actually playable with one hand?

Depends what kind of genre you want. There are lots of adventures that can be played with just one hand...I enjoyed the more recent titles by Wadjeteye Games (e.g. Technobabylon, Shardlight, Resonance, Primordia) quite a lot, they're quite story-driven as well.
Apart from that probably RPGs, especially turn-based ones...if the person for whom that gift is intended likes that genre a good recent title might be Age of Decadence (has a really dark, cynical setting though which might not be for everyone).

Hang on, Robin Hood Legend of Sherwood.

It's great game and only needs a mouse.

Some would say Desperados, but the GOG version is a little buggy.

I've only played about a hour or so of Desperados (didn't like it, seemed like a cheap Commandos clone, and it tended to crash), but is this really playable just with mouse? Sure, it may be possible in theory, but it's a real time game without a pause feature irrc...to play it effectively you might need to utilize keyboard shortcuts. I certainly couldn't play Commandos (which is quite similar to Desperados) just with a mouse, would be an exercise in frustration.

It is fun and also playable with one hand, don't have any use for keyboard in that game. Might require some quick moves during particular sequences though, but still a good suggestion for OP.

Trust me, dt would NOT LIKE HuniePop! XD

Why so? It even has female on female action and female on alien action, no other game is so inclusive (Other than Hatoful Boyfriend) :P

Don't forget the bestiality and the cum shots. :P

This is a woman who gets offended if you don't capitalize the correct letters in her login. And no, that isn't a joke.

Never trust someone that say's trust me especially this person as it was formed by the lies it spews out.

Trust me. I'm a doctor.

WHAT?! There are no great point-and-click games with QTE!

Sorry, I missed the part about "focus on story-driven". That pretty much eliminates most of the ones I mentioned earlier except for Flight of the Amazon Queen (though its "story" is pretty loose and goofy [intentionally ;) ]) and maaaaaybe This War of Mine (in which each survivor you control has a backstory -- assuming you can keep them alive long enough for all of it to be revealed -- and at least a couple possible epilogues for each, depending on what decisions you make during the game).

A strategy title that you may not be aware of is King of Dragon Pass, which fulfills most of your criteria (all, if you consider it being based largely on "choose your own adventure"-type decisions to count as being "story-driven").

Also, most adventure games from Wadjet Eye and other makers of point-and-clicks should be pretty much mouse-only AND story based. Haven't played any of them, though, so I can't give specific recommendations or details.

Wadjeteye games are fine, only one that might be somewhat problematic is Gemini rue because it has some keyboard based shooting sections that could be problematic with just one hand.

Riddles,now?