It's hard to tell by your description what is connected to what and where. Or where specifically you are taking these measurements.
If you take the resistor totally out of the circuit? I'd guess it's measured resistance would be around 0.4MΩ. That's because a 390kΩ resistor is 0.39MΩ.
If you were to connect the resistor in series with the capacitor (observing polarity on the capacitor if its polarized.) Ground the capacitors free end. Connect your volt meter to ground and to the joint between the cap and resistor. Then connect 5V to the free end of the resistor.
Watch the meter. Voltage should slowly rise over about 20 seconds.
If it does, the capacitor us likely good. If not, it may need to be replaced.
Im not sure what it is that makes you think the resistor is being affected by the capacitor. The voltage across is affected, but not the resistor. Read up on RC (resistor/capacitor) networks.
Thanks. I'll try that when I get home. What I don't understand is, as I mentioned above, if I measure the resistance of the resistor when it is in the circuit and the circuit is active, the voltmeter gives me values of 0. When I measure the resistance of other resistors in the circuit, I get the correct values, regardless of whether the circuit is on or not. That's what I find weird.
Anyway, that's an aside. Just thought that commenting on that would help diagnose the problem of why the led won't stop blinking after 20-30 seconds (ever, in fact), once I've pressed the button. I was hoping to get to the bottom of this!