Why does an ice-cube turn faster and faster while melting in water?

The same physics that explains why water tends to rotate when you drain your bathtub also answers this question. In both situations there is a downward flow of water that accentuates the residual rotation that remains after the vessel has been filled. In the case of your bathtub the downward flow results from opening the drain. For the ice cube in hot water the downward flow is created when the just melted water sinks because it is colder than the surrounding water. This downward flow creates a vortex and it is the interaction between the ice cube and the vortex that causes the increase in rotational speed ot the ice cube.

There is an urban legend that claims that the direction of spin in drains is associated with the coriolis effect and differs between the northern and southern hemispheres. While this may be true of hurricanes and cyclones, it is too small an effect to apply to bathtub drains or glasses of hot water with ice cubes. For these smaller scale systems the rotation direction is dictated by the residual rotation created when the vessel was filled. This small residual rotation may not be noticeable until the downward flow accentuates it.

Edit: @sammygerbil has found a website that discusses this phenomenon and attributes it to the same mechanism I outlined above.