To become a mermaid, your sims need to eat mermadic kelp, which you can get in two ways.
1) Your sim needs to be good friends with a mermaid and then use the 'Ask About Fish Parts' interactions while having your sim underwater with your sim's mermaid friend. If the interaction is successful, the mermaid will give your sim mermadic kelp, which you can then eat.
2) Your sim can also buy the 'Mermadic Kelp' lifetime reward for 25000 lifetime happiness points. After that, mermadic kelp will appear on your sim's inventory.
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In the Sims 3 Island Paradise your sims can become a mermaid. In order to become a mermaid, a mermaid must give you the mermaid kelp or in life time rewards you can purchase it. Your sim must be human to become a mermaid (not a vampire, wolf, etc.)
You can't actually become a mermaid but you can download mermaid outfits, that is the closest you can get to being one.
You can't, there is no such thing as a mermaid potion. If you want to become a mermaid on the Sims the only way to do so is by downloading a pre-made mermaid Sim.
They can only have mermaid costumes. They don't look that real you can see their feet. If you want your Sim to be a mermaid, buy the Sims 2 of download a mermaid Sim from the Internet.
The only legal way to download The Sims 3 Island Paradise is to buy it through Origin and then download it from there. Any other way would be considered as pirating and that is illegal.