What could be causing a new 20 amp dedicated circuit supported by a GFCI breaker to trip?
A GFCI breaker only detects ground-faults on the circuit it's protecting. It measures the current flowing on the ungrounded (hot) conductor and the grounded (neutral) conductor, and opens the circuit if there is a large enough difference between the two. A ground fault on another circuit would not trip the breaker, unless it was adding or removing current from the circuit (i.e. it was at fault with this circuit).
If this is a dedicated home run circuit with only a duplex receptacle, there are only 4 places a fault could be occurring.

The breaker.
The wire.
The receptacle.
The device plugged into the receptacle.

The Breaker
Make sure the breaker is connected properly, and that it is properly seated in the panel.  The white grounded (neutral) conductor from the circuit, should be connected to the terminal labeled &quot;LOAD NEUTRAL&quot;.  The black ungrounded (hot) conductor, should be connected to the terminal labeled &quot;LOAD POWER&quot;. The white grounded (neutral) pigtail, should be connected to the grounded (neutral) bus bar in the panel.
It might also be worth replacing the breaker, as they can fail or be defective.
The Wire
If there is a fault anywhere along the length of the wire, this could cause the breaker to trip.  You'll want to test the continuity between the circuit conductors, but you'll first have to isolate them.

Disconnect all circuit wires from the service panel.
Using a multimeter set to measure resistance, or in continuity mode, test the continuity between each combination of wires.

There should be infinite resistance between all conductors. If not, you have a fault.
If you can manage it, inspect the entire length of the cable for damage.  Make sure a staple or nail hasn't accidentally punctured the cable.
The Receptacle
It might be worth it to replace the receptacle, just to rule out the possibility of a defective device.  You'll also want to inspect the wiring, to insure it's wired properly. The black ungrounded (hot) conductor, should be connected to the brass colored screw terminal. The white grounded (neutral) conductor, should be connected to the silver colored screw terminal. The green or bare equipment grounding conductor, should be connected to the green grounding screw terminal.
Also make sure that when the device is mounted in the box, none of the conductors are coming into contact with each other. This is especially true with the bare equipment grounding conductor, as it's easy to create a fault with the bare wire in a cramped box.
The Device
Try to plug in a few different devices, and possibly check the devices themselves for ground-faults.

I can't think of any situations where a ground-fault on a circuit, would cause the breaker to trip on a different circuit. Unless the two circuits are at fault with each other, of course.