Will a bathtub full of water weigh more if I add something that floats in the water?

Mass is mass. If you add something which has mass to the bucket, the bucket now has more mass. It doesn't matter if it was more dense or less dense. If you add 1kg to a bucket, you add 1kg.

Now there are two areas where this falls apart. One is in the case where the object you are adding is actually less dense than the air (not just less dense than the water, but actually less dense than air, like a helium balloon). In this case, we can forget about the minor detail of whether the object is floating on the water or not, we can focus on the entire bucket which is immersed in air. This object displaces a larger mass of air than its own mass, so it will actually add lift to the bucket. The mass of the bucket will still be bucket+water+object, but the bouyancy forces will make the bucket feel lighter. Indeed, this is precisely the mechanism used by hot air balloons.

The other corner case is the case where the bucket is already filled to the brim with water. Adding a single drop of water would cause a drop to have to spill over the brim. Now when we add our toy boat the situation gets a little more complicated. The boat will displace a mass of water equal to the mass of the boat. This displacement will cause the water to rise and spillover. If you tally up all of the masses in this case, you will find that the bucket+water-in-the-bucket+toy boat will have the same mass as the bucket+water did before you added the toy boat. Where did the extra mass go? There's a puddle on the ground outside of the bucket, whose mass is exactly equal to that of the toy boat.

Not only does adding something that floats cause the system to weight more, hanging something that doesn't float from a string and lowering it into the water without letting it touch the bottom or sides of the bucket causes the system to weight more.

Seriously. I do a demonstration of this in class on a regular basis. I put a beaker about 2/3 full of water on a beam balance and adjust the whole until it balances. Then I lower a brass or steel weight in on a string without letting it touch the sides or bottom of the beaker. The scale dips on the beaker side.

There are two ways to understand this:

  • Newtonian reaction If the fluid pushes up in the object (buoyant force), then there is a 3rd Law reaction force on the fluid that is supported by the beaker and in turn by the balance.

    Moreover, you can measure the buoyant force by adjusting the beam weight until it balances once again and subtracting this new weight from the old one.

  • Static pressure When we lowered the weight it it displaced some liquid upward, making the column in the beaker deeper than it had been. The pressure of the water on the bottom consequently rises.

(In both cases I have assumed that the beaker is not full to start in the way that Hot Licks mentions in a comment.)

The bucket is carrying both things: water and boat.

Otherwise I would be able to carry anything in a cup of thick enough liquid (if I could balance it).

The fact that a buoyancy force holds up the boat is no different than when a tray holds up cups and plates with a normal force. Newton's third law says that such force gives an equal force downwards on whatever holds it up. You still feel the weight of both.