HP G60 laptop screen goes on and off with screen hinge movement

If the unit is under warranty return it to the vendor. If it is out of warranty, it is likely there is damage to the wiring harness is damaged or the connectors between the motherboard and the display are loose.
This would require opening the notebook up and checking the cable. Hopefully the connectors are loose and can be tightened and secured with some neutral cure silicone.
http://www.youtube.com/watch?v=MbieJ2UuZnI
Though skip the removal of the hard drive from its carrier and only go up to the part where he removes the front panel and check the connectors leading to the display. If they are loose, this may be the issue. If not, proceed to where he removes the display. Then open the display. 2 screws behind the rubber plugs on bottom of display and unclip the rest. Remove the display panel and check the cable for cracks or loose connections. You may find the cable required for your model on HP Partsurfer, though many of the cables for this model are no longer available, if not available, the cable will need to be repaired or secured to stop flexing with movement binding with tape can sometimes achieve this. If you don't want to attempt this, take it to your nearest repairer.