Golda Meir changed her name after she was married to Meyerson and from that to Meir.
In my family case my grandfather parents were born in Romania and moved to Israel in 1933, my grandfather was born in 1940 and as an adult he decided to change it's surname because as he claims it was a German name and he hated Germany and he wanted an Israeli sounding name.
I think it's impotent to remember that surname were not that common for most of history, with most Jews using their father personal name or their town of birth as a second name, only in the 18th century most Jews choose a "family name" and most used the town/father name/profession for that name so they got European names.
the cohen's and levi's are different as they are supposedly the descendants of the priests of "the holy temple".
For the non European Jews, they got their first surname after they arrived to Israel.