I realize the answer to this probably no, but I'm asking anyway. My computer has a bright blue light on the power button that signals that it is on.

I don't normally care, but I like to watch movies on my computer with the lights off to completely immerse myself in the experience.

Is there anyway to turn this off or no?

I have an HP if that helps at all.

I don't think you can - but I'll leave this question to the computer experts. I do have a suggestion for you tho. Could you cut a small square of black electrician's tape and tape it over the light when you are watching your movie?

Why Not sellotape a colored candy wrapper to filter the light?
For example, A yellow wrapper will turn it to a green light.. worth experimenting with to get a shade which you like

Another Cheapo way is to paste a stamp hinge over the light can buy a big bag of these from a stationers http://en.wikipedia.org/wiki/Stamp_hinge
They are used to affix stamps in stamp albums and they are lightly gummed to avoid ripping the stamp or page

I'm surprised noone has mentioned this obvious answer...

The little blue light is simply a garden-variety LED. If it doesn't have power, it can't glow.

Depending on how they have the wiring set up, it could be as simple as unplugging a tiny connector from the motherboard, or snipping the DC wire going into the LED. Just don't confuse the LED with the power on/off switch, or anything else for that matter.