I have a cheap LCD that sometimes exhibits this problem (Chimei CMV 938D). For me, the most effective solution is playing with the brightness levels - it never whines at 100% brightness. I've always assumed it was the capacitors, though I've never been sufficiently motivated to take it apart and investigate.
It does seems to be temperature dependent - the brightness level at which it starts whining varies day to day, and how long it's been running, so I tend to just leave it on max.
I've had this screen for 2 years now, and it's done it since it was new, so while it might be an early warning of death, I've always just figured it was a function of a cheap build. Admittedly I don't often run mine in the brightness range where it whines.