give us an exact price range
without knowing what you consider cheap, we can't help you
and you mean HD video right?
the term HD doesn't apply to still images, 1080p video is only 2 megapixels, which is horrible for a still camera
in digital photography the term grain doesn't apply either. it is called noise, and is completely different from grain. it also happens on every camera if you set the iso high enough. most cheap cameras will have very similar noise performance, as the sensors are similar in size, and quality.
to avoid noise set a limit on how high the iso goes. most cameras, even if they don't allow you to shoot in manual, will allow you to do that. the problem at that point is that it will lower the shutter speed, thus creating blurry images.
so you can either deal with the noise, or you can deal with the blur.
the thing you should look at if you want less noise, is the quality of the image stabilizer, this will allow you to shoot at a lower iso, and a higher shutter speed, without making the images too dark
