I haven't heard of Mermaids being able to directly turn an ordinary Sim into a Mermaid themselves.
I thought that was just how Vampires did it.
To turn a Sim into a Mermaid you have to eat a special item called "Mermadic Kelp".
This gives a 24 hour Energized moodlet and if that Sim enters the ocean during that time they become a mermaid.
The easiest way to get the special kelp is through the Aspiration Reward Store, it costs 500 reward points, so it's pretty easy to get that way.
Just beware eating two Mermadic Kelps at once. The first will give you the Energized moodlet like normal, but if you eat another one while that moodlet's still there, that's what turns a Mermaid back to a normal Sim.