Nanobots. Given infinite money, presumably a Secret Lab, and some Mad Genius, the best way to secretly annihilate such a species would be self-replicating nanobots. You design them to be able to identify and attack the target species, preferably in a manner that mimics a common disease but cranked up to 11. So say, the nanobots destroy the target's gills and they suffocate, or destroys their nervous system so they die from the various ways that can kill you. Pick any of them. The mimicry is only to throw land governments off the scent until the Deed is Done. You also need to have VERY FIRM parameters for these things. make sure they can only function in water, make sure they won't attack every fish in the sea, make sure they won't replicate themselves into a planet or ocean-wide grey goo. My first thought was to have them only replicate when either A: they are actively attacking a Merperson (hereafter Merson) because they'll probably have to deal with the Mersons' natural defenses like white blood cells. or B: when there are fewer than X number of nanobots per cubit meter of water.
Once the design is perfected you release a vial of nanobots into the ocean. Then you wait. You've done the math, you've calculated replication rates and ocean tides and all the rest of it. You wait until you are 100% certain there is a hundred or thousand or however-many-you've-mathed-out-is-required nanobots per square meter of ocean. Then you hit the ON button. Suddenly every merson in the ocean starts bleeding from the gills and is dead within minutes. Once they've made a kill the nanobots disperse back into the sea. If you didn't mind a bit of lag between pushing the button and Final Victory you could even design the button to spread at a slower rate (maybe triggering every 'bot within 100 miles and then expanding 100 miles a day thereafter) so it looks more like a natural plague to the dirtsiders. That way it'd be REAL hard to figure out exactly what killed the mersons. Even if the governments of the world realized it was nanobots, they'd have a helluva time finding them and purging the entire ocean. Remember you've waited long enough for the whole ocean to be tainted, and they do nothing to fish, people, etc.
Some hibernating mersons may survive this initial purge, but in the long term they're doomed too. The nanobots are in the ocean, dormant unless they either fall beneath the bots-per-gallon level or contact a merson. So every time a merson wakes up from hibernation it breathes its first breath(?) of water, swallows nanobots, and dies within minutes.
And don't forget the final step, popping the cork on some Evil Champaign and kicking your henchman in celebration! Congratulations, you have successfully committed Xenocide!