I have a 17" BENQ monitor that is slow to start after being shut off for a couple of hours. The monitor LED blinks but the monitor takes about 5 minutes to light up. However, if the PC is started up after being shutdown for a short period of time the monitor lights up immediately. I think it is due to a weak capacitor? I'd try another monitor, but I have only the one at this site. Any thought?
The most common cause of what you describe is a failing backlight driver board. You should be able to get and fit a replacement fairly easily!
Having said that, it's not the only possible cause. If you have and know how to use a digital volt meter then you could do some testing to narrow things down.