If a child is small enough, he may not even realize it’s his birthday and the party can be quietly postponed until the following Saturday. If the child is of the age of reason (age seven or so), his parents might discuss with him the appropriateness of offering up having a full-blown birthday party on his birthday in deference to the preeminent importance of honoring Christ’s death on Holy Saturday (while still marking the child’s birthday in some small way with a special dinner, a special dessert, and family gifts).
As a life lesson, children should learn anyway that death always takes precedence over personal pleasure, lest they grow up into selfish adults who refuse to cancel a wedding reception when a close relative dies just before the Big Day while making pious noises that “Mother would have wanted us to go ahead rather than cancel the reception on her account.”