Merseyside Police took a total of 60 calls about the shooting after an appeal on Crimewatch on Wednesday.
A police spokeswoman said the leads were "really encouraging" but said officers would have to investigate further before an arrest could be made.
Officers say they still need to trace a woman seen driving a red car in the area at the time of the murder.
The TV reconstruction of Rhys's murder in Croxteth on 22 August led to a potential witness coming forward.
A woman who was in the area at the time contacted the inquiry team but police have revealed she was not the witness they needed to speak to.
'Further inquiries'
Information about where the gun used to kill Rhys might be found has also been given to officers.
"Twelve people who called in about Rhys's murder all gave the same name but we cannot just go out and arrest that person," said the police spokeswoman.
"It is really encouraging but we need to make further inquiries about why these people have all named the same person before we take any action."
"We can't rest or even try to move on without this person being caught and brought to justice"
A total of 40 people contacted police after the TV programme was aired, and another 20 telephoned the next morning to give information.
One of those who phoned in was the driver of a red car which the alleged killer passed in front of near the time of the attack.
Rhys's parents made the new appeal on the BBC One programme on what would have been the eve of their son's 12th birthday.
The reconstruction featured CCTV footage, enhanced by experts, showing a hooded suspect on a mountain bike.
The cyclist in the footage is seen cycling around the Fir Tree pub, in the moments before and after Rhys was shot in the car park.
Appealing directly to the killer's mother, Rhys's mother Melanie said: "She must know it's him or she must have some suspicion that it's him.
"Even for his own safety she should hand him in. We can't rest or even try to move on without this person being caught and brought to justice. We need it."
Police believe Rhys's killer is about 16 years old and about 5ft 8ins tall.
Merseyside Police have maintained a heavy presence in the area near the Fir Tree pub since the shooting and have made a number of arrests, although no-one has been charged.
Detectives are urging a mystery caller who spoke to the police on 25 August, three days after Rhys was killed, to get back in touch.
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