The raw image file captured by the camera (before it's converted to a jpeg) will be the same regardless of what megapixel setting you use. So changing the megapixel size does not in any way alter the amount of signal (captured light) vs. noise that the camera's processor has to work with when creating the resulting jpeg.

That said, downsampling an image file to a lower resolution CAN result in an image with less apparent noise. This does usually require more sophisticated sampling algorithms than most cameras can easily handle. Even if the camera's processors could handle it, most manufacturers just don't put that much effort into minor features on their compact camera line. So I wouldn't expect the in-camera downsampling when you set a lower megapixel setting to do a good job at reducing apparent noise.

Further, note that this is only true, in the end, for apparent noise. A well-downsampled file will look "cleaner" close up, but will also lose some detail. Viewing prints at equal sizes, they'll be identical at smaller sizes while at larger sizes the full-resolution print will have better detail along with slightly more noise. There's only so much these sampling tricks can do at this point; to really get lower noise, the sensor itself has to be capturing light more efficiently (mostly via better sensel design, as gapless microlenses and other innovations now mean that smaller pixels don't lose significant light relative to larger pixels, making pixel size pretty much irrelevant) or reducing the noise generated by the sensor's circuitry and in the image data pipeline. This is why noise keeps going down, even as pixel counts go up. Sadly, using a lower megapixel setting does not do any of these things.

So to best reduce noise in your images, capture full size with as little compression as possible (or RAW if your camera lets you), then use a program with good noise reduction. There are quite a few available, some with free versions (I generally just run raw in lightroom). After that you can downsample to a lower resolution if you want, but it helps less and potentially reduces detail more as compared to a normal noise-reduction program.