Can You Solve this Ice Diamond Riddle? ft. Vsauce’s Michael Stevens

I'd say the water level goes up, unless you have a quite large diamond compared to the total size of the ice cube.

If the diamond is relatively small, how much the level goes up depends on the relative size of the diamond to the total ice cube how high or how low the total cube will float in the water, but there will always be a little bit of ice sticking out of the water. When all the ice melts, this bit of ice that was above the water will add to the water level. All of the ice and the diamond which was already below the water line when the ice is still frozen won't add to or subtract from the water line once the ice melted.
You could also take into account that a given volume of ice will melt into a smaller volume of water, while the diamond does not change its volume to an amount that will play any role here, but as far as I can see, it wouldn't change that the water level will go up.

However, it could happen that the diamond is quite large compared to the total cube, so that the total density of the cube is larger than water. In this case, the ice-diamond-cube will sink to the bottom of the glass when you drop it in. In this case, melting the ice won't have an effect on the water level except for the reduction in volume from ice to water. So, the water level will go down.

Thus, you can have a net increase or decrease of the water level, depending on the size ratio of the ice and the diamond. If you balance it right, you could also balance these two effects to get no water level change at all.