what is a possible unit for the product VI, where V is the potential difference across a resistor and I is the current through the same resistor?

Recall this equation for a device in a direct current circuit:

P = IV

P is the power dissipated by the device, I is the current through the device, and V is the voltage drop of the device.

If we choose to use the ampere as the unit of current and the volt as the unit of voltage, then the product of the current and the voltage will give the power with watts as the unit.