I live in a two-story house and it's got a single AC unit in a
half-closet upstairs. It can be fairly noisy in the dead of the night
and I was wondering if anyone's successfully managed to minimize the
noise caused by the fan when it comes on. At least I think that most
of the noise comes from it.

The lower half of the closet, below the AC/furnace machinery, is made
up by a box where the duct from the inlet vent downstairs converge in
addition to a vent in front of it. The inside of the box is made up by
naked wood and drywall, a pretty good resonance cavity for the fan
noise. Or so I think, for the upper half of the closet doesn't seem to
make much noise, even when the furnace turns on. Or at least the fan
noise is far louder than the rest.

So I was thinking if putting some absorbent material against the inside
of the box would help deadening the noise significantly. But any
advice is appreciated.

TIA

Augustine writes:

It can be fairly noisy in the dead of the night
and I was wondering if anyone's successfully managed to minimize the
noise caused by the fan when it comes on. At least I think that most
of the noise comes from it.

Your problem may be the fan (squirrel cage) is out of balance from
corrosion. This happens over time due to corrosion from the condensate in
the evaporator. It then vibrates or wobbles a lot and gets very noisy.
Can be replaced, but I rebalanced mine by putting small binder clips
(office supplies) here and there by trial and error.

Cover the walls with something sound absorbant for something quick
ceiling tiles will help, you can contact supliers of sound absorbant
material for more effective control.

Impossible to say since we don't know the cause of the problem. You
seem to want to treat the symptom. The real problem might be something
like bearings that need to be lubed.

Cover the walls with something sound absorbant for something quick
ceiling tiles will help, you can contact supliers of sound absorbant
material for more effective control.

Ceiling tiles inside the duct box is a good idea.

Thanks a bunch.