minhag - Hardboiled eggs and salt water at the Seder

I've participated in only 3 different families' Passover Seders, so my experience is limited in terms of different families' traditions.

How widespread is the custom of eating a hardboiled egg in salt water at the Seder, and what are the source and reason behind it?

EDIT: Is it codified anywhere?

Answer

The Encyclopedia Yehudit suggests the salt water is for the following reasons, though I don't know what the source is for what is written there or if the suggestion offered is their own. This does relate to the reason given for the egg as having to do with the destruction of the Beis HaMikdash - :

1. Salt water symbolizes the tears that were wept over the destruction of the Beis HaMikdash

or

2. It symbolizes the crying of the B'nei Yisrael due to extent of the slavery in Egypt.

or

3. It symbolizes a remembrance of our crossing the Yam Suf (which was salt-water).

את הביצים טובלים במי מלח, המסמלים את הדמעות, שאנו שופכים על חורבן בית המקדש, או זכר לבכיית בני ישראל מרוב העבודה והסבל, או כזכר למעבר בים המלוח.