I was given a Dell monitor (E178FPb) for free, because it quit working. I would like to try to repair it if at all possible but I'm not sure what to look for. When the monitor is plugged in, the LED for the power button blinks rapidly, over and over. Any suggestions? Can it be fixed?

It CAN be fixed... problem is is it worth fixing. And many times its not. Not after you buy schematics, assuming you own or can borrow the needed test gear, and know what to do with all of them... and find out what is actually defective sometimes its cheaper to buy a new one than fix an old one.. Sometimes you get lucky and it's a cheap part that fails.

Okay. Assuming I want to at least TRY to fix it, is there anything specific that I should look for? I saw two swollen capacitors on the mainboard. Should I start by replacing those?

Swollen capacitors might contribute to it but I highly doubt they are a problem by themselves. But without knowing exactly what function they serve in the circuit its hard do say for sure... they are cheap to replace however. Match the capacitance and voltage ratings. As well as the all important type. They aren't all the same nor or they generally interchangible between types.

I just repaired one of these monitors today, which had exactly the same problem. Capacitors C707 and C708 were bulging. These two are output filter capacitors for the +5V supply. The original caps were 680uF, 16V, low-ESR type, rated for 105C. I replaced them with 680uF 25V low-ESR, rated 105C, and the monitor worked perfectly after putting it back together.

I have repaired several DEL LCD monitors of this size and vintage, and by far the most common problem was bad electrolytic capacitors at the power supply output. The bulging tops are a dead giveaway. Electrolytic caps have a limited lifetime --- they "dry out" after a few thousand hours of power applied. And with these power supply designs, they are "always on" at least partially, unless the monitor is unplugged from AC power. When the output caps deteriorate, they start to go low resistance, and drag down the supply voltage. The regulator tries to start up, can't get to full voltage, resets, and tries again, over and over. That's the flashing LED you see at the power button. When the caps go, they usually do not damage any other components, and installing new capacitors will fix the problem.

I've also fixed a few monitors with bad CCFL backlight tubes, but they have different symptoms than yours. I'm 90% sure that replacing the bulging caps will fix your monitor.