Canon EF 35-105mm USM autofocus struggles unless I help the focus ring — what is likely failing?

To start with, this lens was notoriously slow even when working as designed. However, it seems your copy is demonstrating an additional issue which prevents the focusing mechanism from moving properly.

Could some cleaning or greasing of relevant parts of the inside help?

Possibly, but probably not. If the existing lubricants have dried cleaning and relubrication could help, but that's not a common cause for stiff focusing mechanisms.

Or is it likely that the AF motor needs replacement?

Again, not likely. This would be even less likely if it had a ring type USM motor. But in this case the AF motor is the Micro-USM variety. Still, most Micro-USM motor either work or they don't.

Which leads directly to the reason they most often fail - mechanical issues with the actual focusing helicoids that put too much of a load on the motor. This usually manifests itself one of two ways:

• Bent helicoid barrels, particularly on the end that extends out of the front of the lens when focused at close distances. This usually results in part of the focus mechanism's length of travel being stiffer than other parts.
• Worn or broken guide rollers where the inner barrel that contains the focusing elements moves along the slots in the outer helicoid barrel. This causes the focusing mechanism to bind up. Sometimes the screws holding the rollers in place work loose and they fall out.

It's possible the lens sat without being used for a long time. Working the focus mechanism in and out several times may loosen it up a bit. Try to feel if there are any rough spots that are tighter than others along the length of travel. Do this at several different zoom positions. Does the amount of resistance vary or is it always the same?

You can take the lens apart to see if you can find what the issue is and try to fix it. Most people aren't able to reassemble their first lens teardown, but you've got to start somewhere. It might as well be with a $10 lens. Some things to keep in mind:

• Flex cable are fragile. Unplug them carefully and never, ever pull on them. Use tweezers or other clamping tools (surgical clamps actually do pretty well) to grip the connectors on the end of the cable to pull them out of their sockets after pressing any applicable release tabs.
• Some screws are for holding the lens together. Other screws are for holding optical elements in alignment. You need to be able to tell the two types apart.
• Mark the position of everything before you detach it from whatever part it is attached to. Hint: shims and spacers aren't always necessarily symmetrical. They may be thicker on one side than the other.
• Never touch anything if you don't know what it is. Just oil from your skin is enough to ruin a focus position sensor in many lenses.