Get 'em playing Go (aka weiqi or baduk). Idk about open source but there's a good few clients on the Play Store -- BadukPop is probably the most feature rich free one I've found.

Board games in general tend to be less prone to scamminess, though of course aren't completely immune.

Beyond that, maybe grab an emulator for 'em and some old roms. Some games don't fare so well without real buttons (which you can get controllers for) but games like Pokemon where it's not about reaction time work just fine on a touch screen interface.