This is hard(ish). I think it depends on how you plan to use the name, because I can think of some stuff... but I'm not sure how it's being used. I mean, if you're looking for a serious term it'd be one thing, but if you're looking for something really funny then it's another. So I'll give you ideas I have, but they might not all do what you want.
If he plays video games, you could probably summarize most of them as Daily Quests, although vacuuming (as it isn't daily) doesn't fit in that sense. Plus it's not very adventure...y. Still, a lot of games do daily quests that reward you for logging in multiple times, so that is a possibility. It's popular on casual games on Facebook to make people log in frequently by giving them a log in bonus, but big games like WoW have some daily quests that's just taking out a bunch of creatures which I'm sure some casual games do too.. So if he's familiar with it, that might work.
You could give each task a name itself (washing the dishes == cleaning the moat, making the bed == preparing the fields, vaccumming == slaying wild dust bunnies) and give the "challenge" that these would fall under a general name like Requests from the Townspeople or Kingdom Requests. Then he would just need incentive to keep up with the tasks. But this probably wouldn't really work unless you were writing down the chores somewhere. I mean if you're just talking to him, there's no reason to bring up that slaying wild dust bunnies is a kingdom request. Since you already made up a name for vacuuming, you would just say "go slay the wild dust bunnies". Whereas if you were writing this down, you could have it under a header that says "kingdom request" or "request from the townspeople".
But if you wanted to just count all of the chores as one thing, it probably depends on how crazy your naming scheme is already. Personally, I'm with David M on the Slaying Dragons front, because who doesn't want to do things with or around dragons? But if your naming convention isn't that wild (or you already have dragons), War against/on Grime might do it. Then you could call any task that you deem housekeeping a battle if you want to talk about a specific task. For example:
"How did the battle against dust go? Have you been keeping up with the War on Grime?"
But I think David M's answers are really good. I'd use those if they fit your adventure scenario. Especially the obstacle course one! (But the dragon one is clearly the most serious of the bunch.)
Although... if the naming convention is meant to be funny, it might be best just to call it housekeeping. This would work best if the other chores have wildly fantastical names. Like so:
"Today you have to slay dragons, go to the moon, punch all of the titans, and do housekeeping. In no particular order."
Another way to make it funny is to give it a meaninglessly long name with words that don't make sense (ULTRA SUPER FUN MOUNTAIN DELICIOUS HAPPY YUMMY MAGIC SPACE ADVENTURE) as long as the tasks are menial and housekeeping has the longest (and altogether most nonsensical) name. In other words, don't make it a synonym at all.
"Wait, wait. Did you already do the ultra super fun mountain delicious happy yummy magic space adventures? Alright, then you can go out and ride your bike."
You could make your term not a synonym but go another way with the joke by making the tasks sound difficult even though the tasks are easy. But I can't think of an adventure-esque way to do that.