M.U.L.E on the NES is good for some competitive but non-violent fun. Not really exploration based, except for cryst-x finding, but still good. It's semi-turn based, with a simultaneous trading section as the competitve section.
The Heroes of Might and Magic series is a good mix of turn based and exploration, with some pretty cool stuff to do. But there is A LOT to learn, and games can take forever.
I'm not really sure if Machiavelli (Merchant Prince) could be played hotseat. But again, non violent explorey fun, no reflexes required. Heavily trade based. You can hire armies, but you can do all kinds of other stuff too.
There was also a mod / shell for Master of Magic so that you could play a two player hotseat game. Seemed to work fine. Two worlds to explore, and a million ways of doing it.
I'm pretty sure an old shareware game called Capture the Flag could be played as two player hotseat. It's a surprisingly good game in multiplayer.
I'll second Worms as a fun, fairly basic one too.
Things like puzzle-bobble, columns and even tetris are pretty fun to play, either as a high score challenge or as two-player hotseat games. Easy, but hard as well. I actually had a high score challenge going against my mum for a few months there. I'm pretty sure she won in the end.
Card games are also pretty good to play. Most mums can play more than just solitaire. They can also probably beat you at them too.
Anything turn-based strategy on the consoles can work. There's plenty of two player hotseat games. It depends on how complicated you want them. Genghis Khan II comes to mind, but it's complicated as hell until you remember what all the little pictures mean.
Easier shoot'em ups are good for a laugh. Tyrian (or actually, OpenTyrian, it's free now) had a pretty cool two player mode. If your mum wants, she can join your ship and just do turret work if she wants.