I am assuming that the older PC is a full-size Desktop system or tower case. Does the Video connector look like it is coming out of one of the expansion slots or is it part of the motherboard. You can usually tell if it has motherboard integrated video by the location of the connection... if it is right by your other - USB, Mouse/Keyboard, audio, and serial/parallel ports, etc...
If you have two video ports, one motherboard and one expansion... try the other one.
If the video is in an external slot, open the case. Remove and re-seat the video board - it could have a bad connection from being moved.
Then there are the not so obvious... check the cable from the monitor... bent pins?
Does it make the normal single beep during it BIOS diagnostic checks.. or do you hear several beeps?
Does the hard drive sound like it is trying to load the operating system?
Did you hit the side of it really hard with the palm of your hand? - no, don't do that...that's not a proper diagnostic procedure. 
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