If your child is interested in games and computers, there are plenty of free sites you can go to.
This is the place where we usually go to find the links to old favourites and new games, but I'll be adding a few separately as well.
Craymachine is a web toy created by Orange, it isn't really designed for kids, but Emily here likes it. Basically, every letter key makes a funny sound and a little animation/shape, like falling raingdrops, cheeping birds, drums etc. Press Ctrl and + (on number pad) together to make the square bigger (and Ctrl and - afterwards), before you click on it.
Paint like Jackson Pollock is a Flash animation created by Miltos Manetas, and it's incredibly fun and easy. All you have to do is move the mouse around the screen to see different splats appear (the faster you move the smaller they are). Click the left mouse button to change colour, and press Enter to refresh to a clear page. Also, press F11 to get rid of all the toolbars and have a full screen of "canvas" to paint on. Pressing F11 again brings toolbars back.
Few days ago we discovered a new page with games, the Fungooms. It's full of little simple games, but they do require mouse. There are 5 characters, each of them have their own "room" which has about 5 games in it. You can plant, water and pick tomatoes, play a big trombone-like instrument, tickle babies with a duster etc. Emily's favourite is the singing frogs one. Every time you complete a special action (click on all the water lily leaves) or finish the game (pick a basket full of tomatoes), you get a star. The music is nice and not annoying even after quite a long time, IMO. I really like this page. After just a few days, I have noticed that Emily's mouse skills have gone up quite a lot, and she has learnt to deal with frustration. Previously, if she couldn't get or do something straight away, she would start crying and run away, and sulk in a corner. Now she whines a few seconds and tries again. Definitely check this site out. :)
If your kid is very good with a mouse, or a bit older, you could try Octonauts games (they have a few printable colour-in pictures too) or Orisinal, full of beautiful, gentle, mouse-controlled games.
If you have a bit of money to spend, you could try I Spy or Can You See What I See? games. These are based on the book series, and are HOGs, hidden object games, where you have to find a list of items on beautifully set up photo scenes. Every time you click on an item, it will move and make a fun sound. You can get them from Big Fish Games, for example, where you can also download a trial and play free for an hour to see whether you like it. (If you decide to buy, don't forget to join the Game Club, which means you get 30% off every game you buy, plus daily deals etc. When you join, you get billed for one credit straight away, but you can cancel any time - straight after buying the game with your credit, if you wish. Honestly I'm not advertising them here for the sake of it. :)