Rodzaju 21:6 Komentarz: Rashi, Rashbam, Sforno & Radak

יצחק לי means will rejoice on my account. The Midrashic statement is (Genesis Rabbah 53:8): Many barren women were remembered together with her, many sick were healed in that day, many prayers were answered with hers and there was great rejoicing in the world.

צחוק, a joy occasioned by the miracle which she had experienced.

צחוק עשה לי אלקים, even though my son has to endure the pain of the circumcision at such as tender age, nonetheless my heart is full of joy. Therefore, everyone who hears about this event will rejoice on my behalf ignoring the aspect of the circumcision.