The majority of scholars are of the view that zakaah must be paid on the wealth of a minor, if it reaches the minimum threshold (nisaab), when one year has passed. That has been discussed in the answer to question no. 75307.
The nisaab is the equivalent of 85 grams of gold or 595 grams of silver. The wealth of each child should be examined separately, and if it reaches the nisaab then zakaah is due on it, otherwise no zakaah is due.
If the wealth has been invested, as in the case asked about here, then zakaah is due on both the capital and the profits, and must be paid at the end of every year at a rate of one-quarter of one-tenth (2.5%). So if the capital is nine thousand riyals and the profit is one thousand, zakaah must be paid on the total of ten thousand, so 250 riyals must be paid.
This amount (250 riyals, as in the example), may be paid from the capital or from the profits, or from any other wealth owned by the minor.
And Allaah knows best.