Should I tell my child that they have Asperger's?

Be cautious about how giving someone a diagnosis label can inadvertently affect their self-esteem. It can be very beneficial to some individuals to have a diagnosis if they question why they think and do what they do, and for others it can be detrimental should they not want to see themselves as being 'different'.

When they are ready, they will ask.

At the same time, don't hide their diagnosis from them. If they are curious, by all means tell them. Just don't create a problem where there isn't one.