The two boys decided to wrestle for it. Younger brother Chandler had a different idea, however.
"And while they're wrestling, Chandler would come up and eat the cookie and watch them wrestle," Mrs. Jones recalled.
Years later, all three Jones boys are at the top of the sporting food chain.
Arthur Jones, 26, is in his third season with the Baltimore Ravens. The six-foot-three 315-pound defensive end was taken in the fifth round of the 2010 NFL draft.
Jon (Bones) Jones, 25, is the reigning UFC light-heavyweight champion. The six-foot-four 205-pounder will defend his mixed martial arts title against Brazil's Vitor (The Phenom) Belfort in Toronto at UFC 152 on Sept. 22.
And 22-year-old Chandler Jones is a rookie defensive lineman with the New England Patriots. The six-foot-five 260-pound first-round pick made the highlight reel in the Pats' season opener Sunday when he stripped the ball off Tennessee quarterback Jake Locker, allowing teammate Dont'a Hightower to run it in for the touchdown in a 34-13 win over the Titans.
The Jones boys will be front and centre later this month. The day after Jon (16-1 as a pro fighter) defends his UFC title at the Air Canada Centre, the Patriots and Ravens square off at M&T Bank Stadium.
"I'm really stoked about that," said Jon. "I'll be leaving the next morning (after the Toronto fight) bright and early, hopping a flight straight to Baltimore for that."
The star fighter says he will have no split loyalties watching the game. He will just cheer on both defences.
The Jones grew up in a strict household. Father Arthur Sr., 54, is a pastor at the Mount Sinai Church of God in Christ in Binghamton, N.Y.
The kids weren't allowed to stay over at friends' houses or be outside after dark. Sunday was a day for church or Sunday school.
"We did not have time to watch TV because we kept them busy," said Camille.
The Jones parents tried to discourage Jon from fighting for a living.
"I tried to get him to go to Bible college," said Arthur Sr.
"I still think he's taking a long way home," he added when asked what he thinks of his son's career choice now.
Arthur Sr. has also spent time with young offenders and says that helped him how to interact with boys.
"They need a lot of praise and affirmation and redirection. A lot of redirection."
Arthur Sr. ran cross-country, played football and was a Catholic high school champion wrestler as a senior when he went 24-2-1.
Camille was briefly on the swim team as a girl but was never too interested in sports.
Arthur Sr. is six foot one while Camille is a little over 5-8. Her father was 6-7, however, and there is also size on Arthur Sr.'s side of their family.
Born in Rochester, Jon grew up in Endicott, N.Y. The parents say their three boys were very close.
"I think they've always been each others' best friends," said Camille.
"We lived in a rough neighbourhood," added Arthur Sr. "They had to back each other up."
The family also had to survive tragedy together. Daughter Carmen died in 2002 of brain cancer. Camille has had her own share of health issues.
All three boys were good athletes with Camille saying Chandler was good at every sport he tried, from fishing to golf.
Jon says he "sucked" at football, but perhaps only in comparison to his brothers. A skinny defensive end, he didn't start until his senior year in high school when he made the local all-star team.
"I really was a late bloomer," he recalled. "I was kind of starting to get good at it but I never really developed a love for the game because I was so terrible at it no matter how hard I tried."
Plus he loved wrestling, a sport he took up in the seventh grade.
"I lost way more than I won at first, but I stuck with it. I was just passionate at it and ended up becoming really good at wrestling."
He got his nickname at high school.
"Everyone called me Bones because I have really bad chicken legs and I'm a thin guy. I'm really strong, deceptively strong but you can't see it," he explained. "I'm just really skinny."
Camille remembers fondly one year when Arthur and Jon were on the high school varsity football team and Chandler was promoted from the junior varsity squad for the final game.
"I got a picture of all three my boys on the varsity team together," she said.
All three boys were different away from the playing field, according to their parents.
Jon was a scrapper, who always wanted to be the boss.
Arthur was the laid-back, easy-going brother.
And Chandler was an "independent thinker," said Arthur Sr.
"He was the boss but he allowed them to think they were the boss," Camille added with a laugh.
"He was the baby. He out-thought both of them," added Arthur Sr.
A junior college wrestling champion at Iowa Central, Jon was headed to Iowa on a Division 1 scholarship. But his girlfriend became pregnant and the real world beckoned.
Along the way, he found his way to MMA via Team BombSquad, a training centre in Cortland, N.Y. Today he still lives in New York state but refines his skills under the tutelage of Greg Jackson in Albuquerque, N.M.
Jones is now a father of three. Arthur has one child. Camille says Chandler "is planning to just be the uncle."
Jon says he always had high hopes for his brothers and the NFL.
"I knew Arthur would do it. I was hoping Chandler would do it," he said.
"I've come to find out that Chandler, right now, is panning out to be the best athlete in our family. I can say that confidently, man. The kid is just a freak."
Jon also believes Chandler is tailor-made for New England.
"He has the work ethic and just the attitude to be the perfect Patriot. He takes fitness and his job so seriously, at a young age."
"Arthur and the Baltimore Ravens have a great relationship. And they grow closer ever year," Jon added.
Arthur also helps Jon during fight camps, doing everything from carrying his duffle bags to helping him finish workouts when exhaustion was setting in.
"Pushing me, motivating me," said Jon.
In the leadup to Jones' fight with Rashad Evans, Arthur took on the role of Evans — a former Michigan State wrestler who used to train with Jon in Albuquerque.
"He played the role of Rashad, being on top of me," said Jon. "Rashad, back when we trained together, he would hold me down and win that way. And Arthur held me down, with a 300-pound body and gave me more confidence. I knew if I get out from under Arthur, then I could possibly get out from under Rashad Evans."
Evans never managed to take him down and Jon won by unanimous decision.
Since then, Jon has made headlines outside the cage.
There was a drunk driving charge and then a very public dispute with UFC boss Dana White when he refused to fight Chael Soonen on short notice at UFC 151 after Dan Henderson was injured. The card was eventually shelved.
Carmen Jones says people put athletes on a pedestal and don't expect them to make mistakes.
"They're still 24-year-old kids. They still can make mistakes, do stupid stuff, do things that you won't be proud of," she said. "But yet they (the public) forget that they're human beings.
"And so they act once the athlete or movie star or somebody does something, it's almost to them like the end of the world. And I'm thinking 'You know, that's a kid.'"
One mistake for Camille and Arthur Sr. is Jon's tattoos.
"We strongly do not believe in tattoos," said Camille. "He's kind of softened the blow to us with tattoos because one of the tattoos he has is a Bible scripture."
Jon literally wears his faith on his chest, sporting a tattoo that says Philippians 4:13, a reference to the verse "I can do all things through Christ."
Getting the Jones family together isn't easy these days. There's always a football game or a fight.
"It is a big occasion to us when all three boys can be home together," said Camille.
"Last Christmas was the first Christmas we had (together) in a long time," added Arthur Sr.
The family convened in Baltimore at Arthur Jr.'s home for Christmas dinner after a Ravens game.