I recently replaced my old Saginaw power steering pump with a new rebuilt AC Delco unit (which was advertised as having the Saginaw-style reservoir, but didn't). When I drive the car, the power steering seems normal, and the new pump doesn't leak, but it makes quite a bit of noise (especially at start-up). It's not a squealing type of noise, but one that sounds more mechanical (at first, I mistook it for loud lifter or rocker noise). I can hear it with the hood up, but it isn't loud enough to hear over the exhaust (which is pretty loud) when driving.
Any ideas on the cause?

Mechanical noise might be bad. Check for metal in the fluid. What type of fluid is in there? Did you flush it all out or just refill it? Is the box the old one that worked good?

Stupid question but are you sure you are full of fluid? I chased a moaning, gurgling noise on a Chevy pickup once only to find out I was low on fluid.

I didn't flush, I just refilled. I ran the car for a while to get the fluids moving, turned the car off, and then checked the steering fluid dipstick a few times and I didn't see any metal (the pump is full of fluid). The rebuilt pump was sent complete.
The sound that it makes suggests a possible bearing issue. It gets a bit quieter after the engine has been running a minute or two.
A couple of things:
1. The new pump (which was supposed to be a direct replacement, but wasn't) positioned the pulley about 1/2" closer to the front of the car than the old. I checked and the pulley is fully seated on the pump shaft.
2. The rebuilt unit came with two o-rings: one large, and one small. I assumed that they were included in case the high-pressure connection from the old pump needed to be swapped over to the new one. I didn't need use either o-ring.

The pulley could be rubbing or the belt not running true. Worth checking.

Also watch the fluid. A little air in the system can make noise and take a while to work its way out. Also agree watch for metal in the fluid.