This is one of three things. Unfortunately, it's difficult to determine which, unless you start replacing parts.
1. Bad switch. The switch doesn't usually go bad by itself, has the PS2 been dropped? Bumped? Anyone push the switch too hard? It's held in by solder on all 4 corners. If the solder becomes dislodged, the switch can fail.
2. Bad cable. This is rare unless someone opened the PS2 and removed/installed the cable a few times and frayed the internal wires.
3. Bad motherboard. Sometimes a power problem can develop, or the motherboard can short out. It's pretty rare, but I have seen it happen.
4. Bad AC adapter. Recent power surge? It's possible that the AC adapter is delivering enough voltage to power the standby light, but not enough to power the actual system on. You can easily check this by plugged your PS2 up on someone else's AC adapter or taking your AC adapter to someone else's PS2.
You can find parts at the following page:http://store.richspsxparts.com/ve12s.html