I hope that this post isn't too general since it relates all mechanics rather than proposing any specifically. I will try to explain.
The first and comparably least engaging but still enjoyable part of the game for me is combat. Encounters seem to boil down to being strong enough (ie stats, armor, and weapons). The systems involved in combat seem to be complicated at first but in the end I haven't needed to use any other attacks other than punch and chop. The addition of slimes that occasionaly split when attacked is a good example of having to adopt new strategies such as using rapid blows and storm of swords. I feel combat could become more engaging by diversifying strategies based on each mob or trying to counter the strategy of other hearthlings. Each move should be useful in some way and there should be more ways to screw up your opponent if they are doing something predictable. My understanding now is that stats and numbers are everything.
The second and slightly more fun mechanic I enjoy is exploration and finding random rare items. It feels really good to see some glimmermoss (still does even though they aren't rare to the point of non-existence anymore). When I pick a forageable and there's a spider or web it feels awesome. It's fun fishing up something cool or finding it washed up on the beach. (boating around until you hear "Oh Yeeeaaaahh!"). The only thing making these mechanics less engaging is that there is little involvement with the player. While it makes foraging relaxing and meditative, there isn't much to do to involve oneself more in the process.
Currently the most fun I've had with HnH is mining without mine supports. As many of us know, it's a little simplified version of Minesweeper. I think all mechanics of the game would benefit from being infused with this sort of "game within the game" idea. In the example of mining without supports you cannot brute force your way through this mechanic with stats. As a player you have to be aware of the mechanic and plan your actions accordingly.