Assumption: Merfolk possess human anatomy from their waist up with only minor differences, the most significant structural one for hands being webbing between fingers. If this is not the case, my answer is not valid.
Merfolk Need to Get a Grip
Things like hand axes, cord drills, and so on don't actually require all that much hand dexterity to make and use. You can see the hand dexterity requirements for making early tools in this excellent YouTube channel called Primitive Technology. Basically, if you can hold a rock in one hand and make a fire, you can make a lot of things! (Which, in turn, helps you make other things!)
The issue here is a mermaid's grip. In humans, the flaps of muscle/skin between the index finger and thumb needs to stay small or else our grip is compromised: we simply don't have as much strength if we don't wrap our thumb around whatever we are holding! Go ahead and try this: grab a hammer, spoon, or other (blunt) tool, and have someone try to get it out of your hand while using different grips. You, me, and practically everyone else is stronger with the thumb wrapped around.
Mer-people, with a similar hand structure to humans but with a pesky flap, may therefore change how they grip tools. The tools themselves, however, would not dramatically change. There are a variety of ways to hold a tool that can accommodate a webbed hand, such as the thumb running along the length of the tool, rather than around a handle. This can make other kinds of grips, like pistol grips, more popular or even required for mer-folk. As long as this flap is somewhat flexible, I suspect their hands won't be the main limitation on tools.
I suspect they may dispense with handles entirely, opting instead to make their tools easily fit in the palm of their hands, just like ancient hand axes.
Hand-Axes (of the ancient kind), adzes, basalt/flint knives, mortar and pestle, mills, things from cement, and baskets (especially from baleen) are some of the examples of things you can make underwater without much trouble.