can ouput of a 120V AC triac fan speed controller also keep energized a 120VAC relay coil at any setting
This sounds like an interesting project - I'd start by looking at the waveform of the triac to see exactly what you're working with, then identify nearby power sources (or if you're willing to go with battery power, wall wart, etc.).
A triac controller essentially cuts out pieces of an AC waveform. At very low settings, the signal basically is just 'burping' a few volts (depending on the controller I think this can be a pretty low voltage - I'm certainly no expert on all triacs), then being off for the rest of the cycle. So what you need is a way to keep the coil energized - as you said.
If you have access to the full AC waveform (ie before the triac cuts it up), then you can use a transformer to reduce the voltage, rectify to DC &amp; use a relatively simple circuit to watch for those pulses -- depending on how you like to build things, this could be a microcontroller project or an analog circuit build. Since I'm not sure if you have this option, I won't take the time to go into that in detail.
One way to do this if you don't have that access would be to add a rectifier circuit (series diode + capacitor to neutral), rectifying the output of the triac to DC and providing a temporary power reservoir for the coil to draw from. The down side of this is that the higher the setting of the triac, the higher the voltage on the capacitor.
Depending on your relay, this might work, but I would think that it would either (a) only work at higher settings (more voltage), or (b) blow up at higher settings (too much current). To limit power to the relay coil, you can add a constant current circuit or just a resistor.