A digital Leapfrog. They are pretty cheap and way under $1,200. But for real online and multi player games its gonna be extremely hard to find a rely "good" one. But you can find multiple desktop versions under $1,200. Amazon or eBay you can type in "gaming computers". If eBay, there is an option to set the price range. But if amazon, typing in "gaming computers under 1,200" (without dollar sign for better results), will display best results first and you should very easily find what you are looking for within that price But remember that reading the descriptions on listings very carefully is very important. But its hard to say witch one would be the best. It all depends on personal preference.
There are tons of computers that you can buy for even $800 or less that are good for gaming let alone $1,200. Best bet is to build it yourself as it is going to be even cheaper, but if you are not up for that then an $600-$800 computer off the shelf will be able to play any current games.