Firstly, the patty in hamburgers are made from ground beef. So definitely no ham there. But more importantly the term ‘hamburger’ refers to people from Hamburg in Germany (like ‘Londoner’) from whom this dish is supposed to have come from. ‘Burg’ in old German means ‘castle’. The ‘Ham’ in Hamburg is still in dispute. Ham in old German means ‘be crooked’ which could refer to its place along the River Elbe. But the original name was ‘Hammaburg’ in which the meaning and origin of ‘Hamma’ is unclear.
Of course there is no egg in eggplant.
Apparently the name comes from a white egg-shaped variety cultivated by 18th century Europeans.
More interesting is the original name ‘aubergine’ which (according to Concise Oxford Dictionary) has its roots from Arabic and therefore the route (no pun intended, again) through which it made its way into Europe.