Final answer:
The portion of a motherboard's chipset most likely causing a mouse to be unresponsive is the Input/Output (I/O) controller. This component manages communications between the CPU and peripherals. Troubleshooting should involve driver updates and checking for hardware conflicts or port damage.
Step-by-step explanation:
If a user's mouse is not responding and trying a different mouse does not fix the problem, the most likely cause could be an issue with the motherboard’s Input/Output (I/O) controller. The I/O controller is a part of the chipset that manages communication between the central processing unit (CPU) and the external devices such as a mouse or keyboard. It is possible that this part of the chipset might be malfunctioning or the drivers that facilitate communication between the operating system and the hardware could be outdated or corrupted.
To troubleshoot, one should check the device manager for any hardware conflicts, update the relevant drivers, and ensure that the USB or PS/2 ports on the motherboard are not damaged. If the ports are damaged, or if the I/O controller is indeed failing, then the motherboard may need to be repaired or replaced.