I love these ideas. I have a few myself
It wasn't very popular, I don't think, but there was an Australian children's tv show, called "H2O Just Add Water." In the show, mermaids are affected by the full moon and if they look at it or its reflection, they become "moonstruck," unless in the moon pool. This was added clearly for comedic value. Now, for the sims, on a full moon, Merpeople are more powerful. And not only that, but they will want to be in water and will be extremely drawn to the sea. On a full moon, the tides are higher and a mermaid is compelled to swim, and their hydration bar drops a bit faster. But once they've gone for a swim, it fills easily and doesn't drop as quickly.
For powers, if seasons becomes a thing, mermaids can affect the weather. This depends on their emotions:
Happy - Clear the clouds and bring sunshine
Sad - Downpour
Angry - Lightning/Thunderstorm/high winds
Gloomy/Depressed - Fog
Flirty - Sun begins to set slightly earlier, making the sky pinky orange
These weather states can also affect the ocean. So if sims are out sailing and a mermaid is angry, don't sail!
In the fourth film of Pirates of the Caribbean, the mermaid Syrena kisses that Pirate guy and he can breathe underwater. Mersims can give the "kiss of life" to a sim they are romantically involved with or have a high relationship with, enabling them to temporarily breathe underwater without diving equipment.
If it's raining, or a mersim is near a body of water, they a can use water magic. Either freezing sims/objects, soaking things, refilling cups, sinks, etc. Maybe even have water torrent duals with other mer sims to gain skill in Water magic or Mermaid Lore (could be a skill).
Instead of just catching fish like in TS3, maybe they can socialize with them. Maybe add dolphins that mermaids can befriend.
Would be super cool if mermaids could live underwater. Like vampires, they need a specific rest from time to time to replenish energy. Being in water can replenish their energy, the full moon can (it drops at a slower rate), and sleeping on the ocean bed can, too.
Mermaids are also known in mythical lore for their beautiful music. So, maybe add an aquatic instrument only merfolk can play and learn. And they can perform on land or underwater. A harp would be fitting, I think.
To become a mermaid, sims have to find the Pearl of Amphitrite/Poseidon and "observe" it. Once observed, it glows and they become fishy.
I dunno, seems more magical and mysterious than kelp.
And of course, mermaid diet. But this time with actual recipes. They could talk about kelp recipes in TS3, but they couldn't make them. There is a mod for kelp dishes out there, but it messed up my game, so...