A fund has been established to help pay for any gaps in his care, whether it be for experimental medications, or assistance getting around.
His parents on Friday said there were some hopeful signs in his condition.
"His status changes constantly," said his step-father Mark McGowan of Petaluma, adding he is able to breathe on his own. He also has some feeling in his arms.
"They're focusing on getting motor skills in his arms improved. He doesn't have a lot of movement in his hands and fingers," said McGowan.
He was at Cox's side in the intensive care unit of Renown Regional Medical Center in Reno, along with other family members.
Cox's mother, Maureen McGowan, said he was "sitting up" and able to eat. Doctors said he would probably be able to go directly to Kaiser's spinal care rehabilitation facility in Vallejo next week and avoid transfer to another acute care facility first.
Cox, 18, was vacationing with friends Tuesday when he dove into shallow water at a South Lake Tahoe beach and broke his neck.
The accident left him paralyzed from the chest down.
The fund for his rehabilitation will go toward expenses that may not be covered by his Kaiser health plan.
"We know there are going to be rehab expenses coming up that insurance is not going to cover," said Mark McGowan.
He said the money can be used for a wheelchair, van, or other equipment Cox may need to get around.
Cox is a 2010 graduate of Petaluma High School. He was a long jumper and member of the track team.
His parents own Graffiti Restaurant in downtown Petaluma.
Cox had bussed tables there, but planned to attend Santa Rosa Junior College beginning next week. He was interested in a career as a firefighter.
Those plans were cut short by the accident, which still has his family mystified.
Cox said he was running toward the lake and in two to three feet of water, when he did a "flat dive." It was like a belly flop, according to what he told his mother.
Right at that moment, he said a wave created by a boat's wake "pushed in on his head."
While he apparently struck the lake bottom, there was no obvious cut or bruise to his head, according to his family. But he broke his neck and damaged his spinal cord.
Meanhile, Stinger Sports in Petaluma is giving away "Team Cox" t-shirts with a picture of Danny to the first 50 people who donate $100 or more to the fund to benefit Cox.
Donations to the "Danny Cox Rehabilitation Fund" can be delivered or mailed to the First Community Bank, 100 Second St., Petaluma, 94952.