A hard-boiled egg.

The Beitza is to remind us of the Chagigah (Festival Offering) in the Holy Temple

We use two dishes, and Beitza: one as a reminder of the Passover Offering, and one as a reminder of the Chagigah Offering.

According to the Jerusalem Talmud it is customary to use both a Zero'a [literaly arm, or shoulder bone] and an egg - which in Aramaic is called beya [a word which also means "to pray"; "please" ] suggesting "May it please the Merciful God to redeem us with an uplifted arm."

Some texts state that we use an egg, a traditional food of mourning, as we grieve the destruction of out Holy Temple.

This explanation, though, begs further consideration. For it is clear from the Mishnah, and from the Jerusalem Talmud, that even in the times of the Holy Temple they used two dishes. In those days they certainly needed no symbol of mourning.

Alternatively we must say that for hundreds of years they used something else as a reminder for the Chagigah Offering, and after the Temple's destruction they began using an egg.