When it comes to choosing the thickness of wood for a French cleat, there are a few things to consider. First, the weight of the object you're hanging will play a role in how thick your wood should be. Second, the type of wall you're attaching the cleat to will also affect your decision - thicker walls can handle thicker cleats without issue. Finally, if you plan on using screws to attach your cleat to the wall (as opposed to nails), then you'll need to factor that into your decision as well.
Generally speaking, most people opt for 1/2 or 3/4 thick wood when creating a French cleat. This is because it provides a good balance between strength and stability, while still being easy to work with and install. Of course, if you're hanging something particularly heavy (like a large mirror or piece of artwork), then you may want to go with something even thicker - up to 1 thick in some cases. And if you're working with very thin walls (like drywall), then it's best to stick with something on the thinner side as well.
No matter what thickness you choose for your wood, just make sure that both pieces (the one attached to the wall and the one attached to your object) are exactly the same. That way, they'll fit together perfectly and support whatever it is you're hanging up safely and securely.