When an employee changes their name, you are not required under the I-9 regulations to update the Form I-9; however, the U.S. Citizenship and Immigration Services recommends that you maintain correct information on Forms I-9 (note any name changes in Supplement B of form) and keep copies of name changes with the Form I-9.
Employers can't discriminate or retaliate against an employee who updates his/her personal information based on a lawful change of name, Social Security number, or federal employment authorization document.
You can find more information in the Verifying Eligibility section of the HR Library.