A tiny camera by design must contain a tiny sensor. It is the resolution of the sensor which determines how large a print you can make. A 10 megapixel sensor can produce a much larger print than a 3 megapixel sensor can. An old world analogy is comparing a print from a 4x5" negative to that from a 35mm negative.
Technology is capable of producing huge numbers of pixels, but not capable of producing a small cost-effective sensor on which to record them. The result is a tiny sensor with way-too-many pixels crammed onto it. This results in very noisy images above ISO 200, and sometimes above ISO 100. Noise, as we are discussing it , is visible, not audible, noise. It shows up as clumps of pixels, color fringing, loss of contrast, etc. Manufacturers' claims of noise reduction technology are just that, claims. In today's consumer cameras, this technology is ineffective. Also be aware that this is a physics issue. ALL manufacturers currently have their backs against the wall with noisy, small sensors.
What good are all those pixels if the quality of the image is poor?
There are some workarounds. The first is to shoot at lower resolution, for example 5mp. A second option is to shoot only at the slowest ISO, usually 100 on most cameras. This will keep noise to a minimum.
I wish it were this simple. Consumer cameras are used in the automatic modes by most people, which means that the camera selects an appropriate ISO speed depending on light levels. When your child is having a birthday party and the lights are dimmed to enhance the cake candles, the camera automatically boosts the ISO up to 400 or even 800, and you end up with wonderful photos of your family with big clumps of noise all over their faces.
An obvious question is, if the pros don't have noise problems, why should the consumer? The answer is that the sensors inside professional cameras are much larger than those in consumer cameras, and these larger sensors handle noise very well. They also carry a considerably higher cost.
I recommend buying a camera with no more than 5 megapixels of resolution. This will produce prints worthy of framing up to 11 x 14". This should be more than big enough for most of us, who own printers with maximum print sizes of 8.5 x 11". Also, make every attempt to set the ISO to 100 on your camera and leave it there whenever possible. These two solutions will keep the noise to a minimum, and result in prints you can be proud of. Resist the huge megapixel seduction, and save $100 or more on the price of the camera!