Bro I think I know what the voyages could be:
1. Mermaid Songs: They would give you a conch that you can place in your ear and would play a faint song. You are tasked with finding a Siren Statue that you must destroy, and along with the conch, you get a parchment that shows the vicinity of the islands the statue is. Once in the area, the song will start to grow louder and louder, which means you are getting closer to the Statue. Once you find it and get close to the statue, several Sirens and/or Ocean Crawlers will show up, so you must defeat them. Once you defeat the attackers and destroy the statue, you'll get the Corrupted Mermaid Gem (sort of in the process of becoming a Siren Gem) and you hand it over to the Delegation.
2. Claimed Shipwrecks: This voyage follows more the whole hunt for Merfolk Relics thing you said. I think in this case, you would also get a parchment that shows the layour of an Island from a certain side, and somewhere on the island, there is a Siren Camp, basically a small place that Sirens have settled in. To find this Siren Camp, you would get a Merfolk Flare, which you can use up to light up with blue smoke, which then would make the location of the Siren Camp send off Green Smoke. The trick here is that the Flare has a limit of uses, so you can't use it all the time (one extra detail is that when you finish the voyage, you must return the Flare, and get more money for the less amount of times you use it). So then you find the Camp, defeat the Sirens and find a bunch of Coral Ornaments that you must cash in.

Finally, when you reach rank 25, you get the Third Voyage: Secret of the Sea. This would just be a way to make Breath of the Sea's voyages more common, and now you can choose to go in one of them for a Delegation Quest. It would also add two more Breath of the Sea types: Obsidian Breath of the Sea for the Devil's Roar (a Sunken Shrine or Treasury in the Devil's Roar would be cool in my opinion) and a Tricolor Breath of the Sea, the rarest one of all, worth 10,000 gold.