A resistor in series and then a bipolar cap in parallel would make a low pass filter (wired as close to the PID power input as you could get it) that would help reduce the problem if it's voltage sag. The only issue would be making sure the filter's cut off frequency is at least 10% above 60hz so you dont damp the main AC supply.
The filter will work the same on DC as AC if you have it set up right. The filer would effectively "short" any higher frequency pull to ground, which is exactly what you want, and the resistor would act as a damper to make sure you dont run too much current, while not acting as a "short" for lower frequencies.
The only real load the filter would show to any load wired further down the AC line would be a phase difference, possibly changing the complex part of the impedance, which for a solenoid shouldn't make a big difference, if you had a 3 phase motor down the line I might worry, but a standard single phase motor like your pump, it shouldn't effect. (I'm assuming the filter would be wired only to the PID and that the PID is only switching, and not supplying the power for the solenoid).
The MOV is a good idea, because for an AC circuit it acts similar to what is called a "free wheeling diode" which will prevent the inductor in the solenoid from causing a voltage spike when it's current is shut off. The voltage spike from the inductor comes at a negative voltage from the original voltage direction at the time of disconnect, thus at DC placing a diode backwards across the coil allows the spike current to run backwards through the coil dissipating the magnetic field quickly and removing the voltage spike from the system. The MOV is the same idea but for an AC system.
The filter circuit would also reduce the impact of a voltage spike without the MOV. The filter only cares about the frequency of the spike, up or down in voltage, the MOV will only protect against an upward "absolute" upward spike in voltage. (aka if you're at -60 in your AC cycle and the spike took you to -160 very quickly, or 60 to 160 quickly, the MOV wont protect against -160 to -60 or 160 to 60) the filer circuit would reduce the effect of both, but the MOV is much better at preventing the absolute spike.
Now, an inductor acts to current as a capacitor does to voltage. Thus you never have a fast change in current over time across an inductor. Based on this, I'm going to bet that your issue is most like a voltage spike, rather than a voltage dip. Turning on a compressor motor usually causes a big load spike because at 0 RPM the motor uses the most current and has the most torque, but the instant the switch is turned on the current pull is low until the coils saturate and then pull large current. (goofy transient response I know).
but bjornbrewer (/wave at a fellow EE) is right:
Another method for suppressing voltage spikes is the transient voltage suppression diode (TVD). Although diodes do not have as much capacity to conduct large surges as MOVs, diodes are not degraded by smaller surges and can be implemented with a lower "clamping voltage". MOVs degrade from repeated exposure to surges and generally have a higher "clamping voltage" so that leakage does not degrade the MOV. Both types are available over a wide range of voltages. MOV tends to be more suitable for higher voltages, because they can conduct the higher associated energies at less cost.[2]
thus using a TVD may be your better solution:
http://en.wikipedia.org/wiki/Transient_voltage_suppression_diode
in this case, you'd want a bidirectional TVD with a breakdown voltage of about (120 * root(2) for RMS) 170v, and be able to handle a good surge amperage.
If you're really worried about the life of the MOV get a high joule ratted one or write several of them in parallel (as suggested on the wiki page).
As Bjorn said, if the problem is at turn on and causing a low voltage, then the problem is most likely a high impedance in the supply wire, beefing it up may solve the problem, and making it shorter will help too, or you could run the solinoids on completely different circuits from the PID using the PID to control a switch that then turns the power on/off for the solenoid. I could see this as the case if the solenoid is having trouble closing and pulling a lot of current once the inductor is charged but the solenoid is not moving (like a compressor motor). Most compressor motors have a big cap in parallel before the motor to smooth out the voltage drop when it first turns on.
The real question is, does it have its problem when the solenoid is shut off, or turned on? If you can figure out a way to answer that question, then the solutions are in all the posts above.
-devon BSEE
-free wheeling diode:
http://en.wikipedia.org/wiki/Free_Wheeling_Diode
- MOV:
http://en.wikipedia.org/wiki/Varistor