Mixed signal design is a lot easier if you have a good ground plane.
I think you have some EMC problem here. Basically your desktop chassis, the signal generator's chassis, and the scope's ground, are all grounded (to safety ground), but the laptop is not grounded.
Some common mode current will always want to flow between whatever is plugged at the input of your circuit and the computer case. This current can come from transformer interwinding capacitance, antenna effects, etc.
If both ends are grounded (desktop chassis and signal generator), no problem, the current will flow in the ground wires.
With an ungrounded source (laptop), common mode current must flow in your device's ground.
A big fat ground plane has an extremely small impedance, so there would be no problem except at RF, but the common mode ferrites take care of this.
But you have skinny long ground traces (I can't even locate which one is ground) which are high impedance, thus you'll get nasty parasitic voltages in your ground. Basically all the "GND" voltages in your circuit become random, which is not so cool.
When you connect the scope's chassis to the thing, the scope ground (via a fat coax shield) becomes the path of least impedance, and it sinks the common mode current. So it works.
The fact that it works when you put the ground clip on the USB connector makes me think that the laptop emits the offending current, probably due to the interwinding capacitance in the laptop's power brick.
Advice :
As a quick fix to this design, just take a fat piece of wire (lampcord will do) and connect your circuit's INPUT GND to the USB shield. Make the wire as short as possible (ie, straight).
The USB spec says you're not supposed to do that, but noone's going to sue you anyway. Also consider clipping a ferrite on the USB cable, so your circuit doesn't become a radio receiver.
Next time, read up on EMI/EMC, use a nice ground plane, put some ferrites, etc. If you want input and laptop grounds to be separate, use optical isolation, or at least a differential input amplifier with good CMRR.