Sherry West, whose son Antonio was shot in the head around 9 a.m. EDT Thursday, told police two African-American youths approached her and demanded money, CNN reported.
West said one of the boys was 13-15 years old and the other may have been as young as 10.
She said one of the attackers told her, "I'm gonna kill you if you don't give me your money."
She said the attacker shot her in the leg when she said she had no money.
"I put my arms over my baby and he shoves me and he shot my baby right in the head," West said.
West said the two attackers ran into a residential neighborhood.
Brunswick Police Department spokesman Todd Rhodes said investigators were "not ruling anyone or anything out."
Rhodes said Antonio was shot with a handgun but no weapon has been recovered.
"However, let me assure you -- as I previously indicated -- that we are thoroughly investigating this case and we will not, I repeat, will not leave any stone unturned," Rhodes said.
He said West was cooperating with investigators.
Antonio's father, Louis Santiago, told CNN affiliate WAWS-TV, Jacksonville, Fla., the attackers "should have just taken the pocketbook" and gone on their way.