The mother, Sherry West, told WAWS-TV that two boys approached her and demanded money Thursday morning while she was walking near her home in Brunswick, about 80 miles south of Savannah.
West said she insisted she didn't have any money and tried to protect her son, Antonio, before one of the boys opened fire.
"I put my arms over my baby and he shoves me, and then he shot my baby right in the head," West said. She was shot in the leg.
Police spokesman Todd Rhodes gave few details about the investigation Friday, but said no weapon has been found and that investigators were checking school records for leads.
Several people in the neighborhood called 911 after they heard the gunshots fired, but Rhodes said investigators believed that the mother was the only witness to what happened. Rhodes described the neighborhood as safe
"Understand this: There is no clear motive right now," he said.
He urged anyone with information to call Brunswick police. There is a $10,000 reward for information leading to an arrest and conviction.
Officers from a SWAT team checked vacant houses as investigators tried to find possible witnesses. The Georgia Department of Natural Resources provided a helicopter for the search. A sketch artist from the Georgia Bureau of Investigation was being sent to Brunswick.
The mother lives in a rented house in the city's Old Town historic district. Beverly Anderson's husband owns the property and she said West has lived there for six or seven years. Anderson said she spoke with her yesterday to extend her condolences and see if there was anything she could do for her.
"We're just very sorry about what happened and very aghast that something could happen in our little neighborhood," Anderson said. "It's a quiet, safe little neighborhood."
Anderson said people walk up and down the street, children walk to school and families are frequently outdoors. "It's scared everybody," she said. "They don't feel so safe outside."
She said West stayed home to care for her baby, who was often spotted in his mother's arms.
"The house has a front porch with a swing and we'd see him out on the swing with his mother," Anderson said. "He was a happy, cheerful baby."
Associated Press Writer Christina A. Cassidy in Atlanta and news researcher Monika Mathur in New York contributed to this report.