It took my diver several tries before a mermaid unlocked it for her. She was friends with both Salty Seaworth and Maya Ocean. She lived on a houseboat parked over the Mermaid Cove diving spot. She'd summon both mermaids to her boat, then dive with both of them. Most of the time Salty would constantly talk to her about fish, sharks, kelp recipes, etc. Sometimes she'd ask them about fish parts on her own, and wound up with several pieces of mermadic kelp. I never told her to do anything once she dove, though I would occasionally move her away from sharks or cancel out any random diver's attempts to talk to her.
After several dives Maya finally decided to show her the mermaid island, but even that didn't go as planned - because they were in a group all three of them went, and they wound up taking a water taxi to get there. At least the island finally unlocked!
Just keep trying it with different mermaids or as many mermaids as you can get in a group. I have a theory the original four (Salty, Triton, Maya, and that one who looks like Ariel) are most likely to show a sim to the island.
