The kid's games... someone I think suggested running and that sounds good. Also, they have been practising archery the last few weeks so they should show people what they have learned. But no wrestling, please. I think they have had enough of that already...
One possible event could be that old game where two people face each other standing on a log that has ben hoisted up (like a meter or something) and they have heavy sacks as weapons trying to stay on the log and make the other fall. I can't remember the English name for it but I hope you know the sport. In that there might be a female competition as well.
Maybe something like dagger-throwing for the women or something? I mean at least I think the rohanian society was quite a traditional one in respect to male and female roles - with possible exceptions like that of Eowyn - and so the female "games" should be less dangerous and possibly less physical or aiming less to physical damage in the first place.
How about a competition which would include different tasks to perform? I mean tasks that would require agility, firmness of hand, wit and such. It might even be a game for all (also males and kids) as there might be interesting results. The tasks could be something like climbing to get a required object, pebble-throwing, building from pieces of wood a tower of a required height so that it doesn't collapse, figuring out any sort of puzzles, running/crawling/jumping through a track with different obstacles etc.
Or something like that.