The name יצחק means he will rejoice, meaning that this will be shown to be good. The term used for Yishmael, מצחק, means that he mocked actively and viciously. It therefore is a completely different usage.
Rav Hirsch explains that their are two different meanings involved. Vayeirah 21:3
3 And Abraham named his son who had been born to him, whom Sarah had
borne to him, Isaac.
the birth of this child was צחק, something, according to natural
conditions, rather laughable (see above on Chapter XVII,
So when Abraham called his son יצחק these facts were vividly in his
The statement in Vayeira 17:17 shows:
17 And Abraham fell on his face and rejoiced, and he said to himself,
"Will [a child] be born to one who is a hundred years old, and will
Sarah, who is ninety years old, give birth?"
On the other hand, Yishmael actually went to great extremes in his making fun of the ideas of Avraham. As Rav Hirsch says on Vayeira 21:9:
He took in just as much of the great ideas of Abraham to make him
ironically disdain them, and over that which the world in general
greeted merely with צחוק, in which (V.6) derision was only lightly
mixed, Ishmael was completely מְצַחֵק, and was therefore completely
unfit to join with Isaac in the heritage of Abraham.