Abox sliding on a frictionless surface when it suddenly begins to rain heavily. if the downpour of rainfall falls vertically into the box, gradually filling it, what would happen to the speed of the box

The speed of the box will decrease and the box will slow down.

Initially the box with some mass is moving with some speed. hence we can say the box has some momentum. The momentum of the box must remain constant. as the box gains water due to downpour of rainfall, the mass of the box or system  increases . momentum which is product of mass and velocity has to remain constant. To remains constant , as the mass increase the speed decrease so as to keep the product constant.

see earlier answer