The work Yalkut Yosef (English edition, p. 262) records a custom to eat a boiled egg immediately after Kiddush at the Seder, in commemoration of the Korban Hagiga, a sacrifice which was brought along with the Korban Pesah in the times of the Bet Ha’mikdash. This is not, however, the custom we follow. We eat the egg just before the meal, after we complete Maggid and eat the Masa, Marror and Korech. This way, we avoid the Halachic issues concerning the quantity we are allowed to eat and the recitation of Beracha Aharona that would arise if we would eat an egg after Kiddush.
Many commentators addressed the question of why an egg was chosen as the means by which we commemorate the Hagiga offering. Clearly, no eggs were ever brought as a sacrifice. Why do we commemorate this sacrifice with an egg?
Some scholars suggested that an egg is used for this purpose because it is the food traditionally fed to mourners, Heaven forbid. The reason why we cannot bring the holiday sacrifices is because the Bet Ha’mikdash was destroyed, and we thus eat an egg, the symbol of mourning, to remind us of the tragedy of the Temple’s destruction, as a result of which our Pesah celebration is incomplete. (For this same reason, the first night of Pesah always falls on the same night in the week as Tisha B’Ab. This year – 2013/5773 – for example, the first night of Pesah is a Monday night, and if we must still observe Tisha B’Ab this summer, it will be observed on a Monday night. On the night of Pesah we bring to mind the tragic events of Tisha B’Ab, as a result of which we cannot offer the Pesah sacrifices.)
Others explain that the Aramaic word for egg is "Be’a," which also means "wish." We thus eat an egg at the Seder to express the fact that G-d sincerely wishes to have compassion on us and redeem us ("Ba’eh Rahamim Alenu").
Another explanation is that an egg has no opening, and thus symbolically represents somebody whose mouth is "closed" and is unable to speak. The miracles G-d performed in Egypt had the effect of "closing the mouths" of those who denied the possibility of Beneh Yisrael’s redemption, and we therefore eat an egg to commemorate this important effect of the miracles of the Exodus.
The Hatam Sofer (Rabbi Moshe Sofer of Pressburg, 1762-1839) suggested that unlike other foods, which become softer the more they are cooked, an egg becomes stiffer the longer it stays on the fire. It thus symbolizes the Jewish people, who become stronger as a result of the oppression we endure. Persecution has not weakened the Jewish people, and has only made us a stronger and more confident nation.
Yet another explanation is that an egg has two "births," so-to-speak. It emerges from the hen, and then the egg cracks and the chick emerges. The egg at the Seder thus symbolizes the dual redemption which our ancestors experienced. First, they were freed from Egyptian slavery, and then, seven weeks later, they received the Torah at Mount Sinai, which marked the second stage of our redemption.