Yes, downsizing an image greatly reduces noise. It's the same concept as discussed in the tutorial on Noise Reduction by Image Averaging. If you reduce the total number of megapixels by 50%, such as 10MP to 5MP as mentioned, you will reduce the amplitude of noise fluctuations just as with averaging two photos -- except in your case you'll be sacrificing resolution as opposed to exposure time and/or storage space.
For those interested: the amplitude of noise drops proportional to the one over the square root of the number of photos averaged. It's the same with pixel averaging when downsizing. If the original resolution is 10MP and you downsize to 2.5MP, then the amplitude of noise will generally be cut in half. This is because the number of pixels are reduced by a factor of four (and therefore 1/[square root of 4] = 1/2)
As to whether it is better to shoot in low resolution mode or to downsize afterwards: it really comes down to the interpolation algorithm used in your camera versus the one you intend to use on your computer. Back in the day (ok...just 5-8 years ago or so), in-built digital camera downsizing often took shortcuts to save on processing power, which usually corresponded to an inferior downsizing result. This would include either bilinear interpolation...or in a worst case scenario, some cameras would not even use all the original pixels, and would instead perform something similar to a nearest neighbor interpolation (which is like just picking every other pixel and discarding the rest of the image information).
There's a lot more on image downsizing here: Image Resizing for the Web & Email
In addition to a discussion of interpolation algorithms here: Digital Image Interpolation
Something to emphasize is that even if you reduce noise by downsizing the image, you are only doing so for images which are viewed at 100% on-screen. If your image has to be printed large enough that even the full resolution 10MP original is just barely sufficient (such as from 8x10 to 11x14 inches in size), then decreasing noise by downsizing will do you no good, and can in fact make the image much worse.
One last thing: the above discussion does not apply to half size image that are created by cropping out the central portion of the frame, such as with some Nikon models. In this case, noise will be identical at 100%, and worse when printed, since the digital file needs to be enlarged more for a given print size.