Hi there Steve and Welcome!
The "bright individual pixels" you describe are called "hot pixels". (I'm assuming they aren't dead pixels since you state they only showed up on your recent series of dark, available light, long exposure shots.) Hot pixels typically look like bright white dots sometimes fringed with a bit of color. These can occur on long exposure shots taken with a digital camera. You may also sometimes find them in high ISO shots as well. These pixels are individual sensors on the CCD that have a higher than normal rate of charge leakage. Hot pixels differ from standard noise which looks more like anything from film grain to random dots or bits of color. This is not unusual for digital cameras.
Some digital cameras have a built-in "noise reduction" mode which reduces the appearance of these excited pixels. The default setting is usually OFF; next time, you'll want to turn it ON. But this only helps pre-processing not post-processing.

Besides cloning out the excited pixels in post-processing, there are software filters designed to eliminate them in photo editing. One such post-processing filter is HotPixels Eliminator (
http://www.mediachance.com/digicam/hotpixels.htm). I don't have first-hand experience with it but it's an example of what is available. Pixel Zap is another (
http://www.tawbaware.com/pixelzap.htm). I'm sure there are lots more options.

Hope this helps,
Deb