Suggestion: Pugilism
So, Bannerlord's fist fighting is much better than in the previous titles, and I think it could use some love.
A fist fighting arena, which would work like tournaments. allowing only 1v1 combat would fit that, the rewards being mostly money and/or relation gained with people who bet on you. This system could work as an addition to the criminal system we got in Bannerlord. And could serve as yet another way to up your relations with gang members, merchants etc, just the notables. Or just make it a casual, ancient greek era beat down where every contestant is naked.
EDIT: Also to spice up the fist even more, let us shove/push people using the block+kick combination while unarmed.