How can I test the integrity of my neutral wire?
The problem could be either (or both) a marginal neutral wire and/or an unusual load on the circuit.

To narrow it down, I would place an oscilloscope at a representative point on the circuit&mdash;such as an outlet beyond the lighting.  With a dual channel scope, probe the hot and neutral wires with the scope grounded to the circuit ground at the same point.  That way the scope will see the same conditions as the lighting.

Experiment with various sweep frequencies at a time the lights are flickering.  On the hot wire, square waves would tend to indicate an intermittent wire connection.  Spikes with harmonics in the 100 kHz to 5 MHz range would indicate arcing which could be a almost-broken wire or loose connection.

A significant signal (> 0.5 volts) on the neutral probably means the wire is not well connected.  With an assistant, see if wiggling outlets and switches affects the signal.  Or simply shut off power and open every outlet and switch and tighten the wire connections.  Also, open the breaker box and firmly screw down every neutral and ground wire in the terminal strips.  If that doesn't clean up the neutral, unplug/turn-off everything else from the circuit to see if there is any effect.  1980s-era switching power supplies (such as in computers) could cause this problem.

Follow up with a comment about any findings so someone (maybe me) can help pinpoint what they mean.