Kids "learner" boards, much like an adult starting out, need to have enough volume to both catch a wave and then have the stability in order to stand up. The sizing depends on their age and size.
These boards need to be long, wide and thick, relative to the riders size. Most people will start by hiring a 'soft top' (see soft tops) that are particularly user friendly as they have a foam deck and dont hurt! These boards are ideal for catching waves and offer the most stability when trying to stand up. There are many learner boards on the market such as NSP's and Pop Outs that are injection molded plastic boards.
These are usually robust and hard wearing but tend to be a lot heavier than a custom board (made from foam and fibre glass). This can cause problems with smaller children trying to surf as they'll physically struggle with these weightier boards.. Some times a 'foam and fibreglass' board is the solution. However you'll notice that the 'plastic' boards tend to be cheaper.
Whatever type of board you go for, be it a new custom board or a molded board, your board will need to be of the same sort of size. Kids learner boards are usually between 6'6" and 7'6". The easiest way to look at it, is to imagine that a 7'0" is ideal for an average 12 year old.
This isn't to say a child cant surf on something smaller or something bigger either. There's a good chance a child will be able to stand up on something smaller, but then will struggle to catch waves on their own. The problem going too much bigger is that they wont be able to physically handle the board in the water.