This is a good question.
字了一 should be understood as:
Thousands years ago, many people in China have a special name besides their first name and last name though nowadays most of Chinese don't have one. And this special name is often described after '字'. For example:
According to some reference in Chinese, 字 sometimes can be changed. For example,
关羽 was using
长生 as his 字 but then changed to
云长 (which is mostly known by all Chinese)
字 can be given by a person himself/herself or his/her friends, mentors. While ordinary names were mostly given by the parents.