You can hear this noise when the blower is on, kind of a click/gurgle. I've burped the cooling system thinking there was air, but it's still there.

It almost sounds like air making something flap open and closed. All the vents work everything functions normally, and the noise goes away after it's been on for 30 seconds to a minute. You can kind of hear from the video, but the noise speeds up as the blower setting is moved from low up to high.

Thoughts?

http://www.youtube.com/watch?v=2nwaqFFtlBI

 

if you look behind the pass side front wheel liner you will see the two heater core hoses.

both are put together with a coupler inside the hoses and held together with clamps.

I would go to lowes and look in the plumbing section and buy a 5/8 or 3/4 barbed valve. take apart either one of the heater core hoses and remove the coupler and put in the barbed valve and shut it off.

then see if your issues goes away. i have one on my truck and not only will this likely tell you if the heater core is making the noise but shutting off the hot water in the summer will net you cooler ac in the summer

 

maybe it's the recirculation door inside the HVAC unit... It could be partially damaged and close to falling off. That's a really common problem, and I'm pretty sure that 'door' is right above the blower motor. Or it could be sticking because the gear splines are worn out and the motor is trying to get the door in the correct position.
After the noise goes away, have you tried changing the position of all the vents (floor, main vents, to defrost) and changed from recirc mode to fresh air mode to see if that noise will start up again?

** That's a cool trick Weed, and it will produce cooler AC in the summer, Hell yeah! Add that to my list