McNery died April 30 from complications of cancer. He had served as municipal court judge for 37 years.
McGraw served as Round Rock mayor for three terms, starting in 2008.
The council also appointed Lucas Wilson as a new associate municipal judge. Wilson is now the city's third associate municipal judge.
Round Rock Municipal Court, 301 W. Bagdad Ave. Ste. 120, Round Rock. 512-218-5480. www.roundrocktexas.gov/departments/municipal-court