Blower motor resistor

The most likely suspect is the fan motor itself, but if it has tight bearings, it will draw excessive current and burn open the thermal fuse on the resistor assembly. That can damage the new resistor too.

It is also possible the motor simply has worn brushes, especially if it is used on the higher speeds most of the time. For that, you can often get them started by pounding on or near them while the ignition switch is turned on.

The next place to start is by measuring the voltages on the motor's wires when it should be running. To be accurate, that should be done with the motor plugged in. On most cars you can back-probe through the back of the connector to take the readings.