I'm having the same problem. I've just replaced caps to fix the click-on-off problem and now have a dark picture. (LN-T4661FX/XAC)
Hopefully I can add a bit more info...
I had the click-on-off problem for quite a long time - I just put up with it. One day when it came on (eventually) it had a dark picture (mostly blue). I was hoping it was related to the bad caps and figured it was about time to replace them. I did just that (replace all three budged 10v 1000uF caps).
It powered up right away (first time in a long time) but the picture was still dark. I tried multiple sources and cables and narrowed it down to all three HDMI inputs - analog inputs seem to be okay.
I'm a rookie at this stuff but what I'm guessing is that since I ran with bad caps for so long, other damage was done... where it is just the HDMI inputs that are affected makes me think it's the main board (or potentially still a problem with the power board) but it's not likely the t-con board or anything else down the line.
It looks like I can get a main board for about $100 on ebay - I think I might give it a try - but I'd be more comfortable making the $100 gamble if I had more info.
Has anyone solved this one?