PC wont turn on. How to find out if PSU, motherboard, or CPU is bad?

I don't know whether this is the right category or not but I am trying to see if a computer is worth throwing money into to find out what is wrong with it. This is not my computer but I was asked to try to find out what is wrong with it as it will not turn on.
The computer was bought around 2 years ago and now the monitor will not display anything? The computer is:
HP Pavillion a6500f
Pentium Dual Core E2220
4GB memory
500GB Hard Drive
NVIDIA GeForce 7100
250W Power Supply
Windows vista Home Premium 64 bit

(non-factory components) both installed about 1 year ago
VisionTek Radeon HD 3450 Graphics
dynex wireless card

I dont know exactly when it started not working properly as it isn't mine. No hardware was recently installed inside the case (perhaps usb drives or wireless mouse/keyboard etc) but to my knowledge not around the time it started not working. I was told that the screen went blue and it turned off. Then it wouldn't turn back on. She tried again the next day and it turned on, booted up and started windows normally... after a couple minutes the monitor went dark and couldn't do anything but hold the power button down to shut it off. She started it up and it did the same thing one more time. Now when you turn it on it wont display anything.

I have tried starting it up normally, reseating components/cables inside case, cleaning dust out from inside case (found alot of dust around cpu. I turned the computer on with case open to see what was working. The case fan, cpu fan, psu fan all work. I hear the hard drive spinning, dvd drive opens and closes. I have worked through a couple different strategies people suggested to troubleshoot similar problems. Clearing CMOS (by removing battery and switching jumper (together and independently)), swapping memory bays, using 1 stick instead of both (tried both sticks in both slots individually) disconnected everything except cpu,cpu fan, psu, and 1 stick. Even tried without any memory sticks just to try to hear a beep. Tried a similar 250w psu from an older working computer. I even tried to take everything out of the case and set it on a piece of plywood as I seen on this site (breadboarding, i think it was called) Still have not heard a beep with any of my tests.

I was wondering what my next step should be. I dont want to have her put a lot of money into it as she might can get a newer pc around black friday for what it may cost her putting into it. Is the motherboard officially dead? I seen on amazon reviews looking for solutions that 7 of the 13 reviews claim motherboard issues. Could it be the cpu since it was covered with so much dust. Or is the power supply gone/not strong enough? But why wouldn't the other old 250W work to at least turn it on once as the one in it has worked for 1 year with the same components?

Sorry for the long post, but i wanted to make sure I covered everything and I wanted to make sure I dont tell her she needs a new computer if it is something simple that I overlooked? And I dont want to tell her it will work if she buys (psu,cpu,motherboard) if it may not work. But if I know it will work then it will save her a lot of money. I am open to any/all suggestions.