Alfie Dingley, aged six, suffers from a rare condition and his family say he needs cannabis oil to help reduce his seizures.
He said: "How could one not support Alfie? Hearing what his life has been and the benefits given to him by being able to use medicinal marijuana.
"There has never been a stronger case for the legalisation of medical marijuana.
"I have been registered for medical marijuana in California for over three years and have found it immensely beneficial for my arthritis.
"I had to have eight steroid injections in my fingers and knuckles, which was about as painful as anything one could imagine, because medicinal cannabis is not available here."
Ms Deacon said the meeting was a success and ministers have approved in principle Alfie being issued with a special licence for the medicine.
"We had a positive meeting, they accept the compassionate issuing of a licence for Alfie," she added.
"Now we need our medical professionals to write the prescription and they will expedite the licence for him."
Alfie, from Kenilworth in Warwickshire, can have up to 150 potentially life threatening seizures a month.
He has been taken to hospital three times since returning from the Netherlands four weeks ago, where his symptoms improved after he was treated with cannabis oil.
A number of other high-profile figures have backed the campaign, including Joanna Lumley and Sir Richard Branson.
Mr Dingley described the support as "amazing" and added: "Alfie's condition is worsening, which is obviously a worry.
"The steroids have side effects, they make people more aggressive and we've seen a change in his behaviour.
"We just want our little boy back, our happy little six-year-old playing with his sister."
He added: "This is our six-year-old son, we're not going to put something into him that's in any way illegal.
"What we're asking for is a medical grade product, made under laboratory conditions, which is bottled and prescribed in the way any pain killer is."
Mrs May's official spokesman said the PM had previously written to the Dingley family to express her sympathies and "reiterate the Government's commitment to explore a range of options to find a solution for Alfie".
The spokesman added: "That work is ongoing and is being led by senior clinicians.
"What we have said is that it is important that medicines are thoroughly tested to ensure they meet rigorous standards before being placed on the market, so that doctors and patients are assured of their quality and safety."
The spokesman said that Mrs May was opposed to the decriminalisation of cannabis.
Asked whether decriminalisation was one of the options under consideration in the current work, he said: "We have said when we've looked at this in the past that we are looking at specific solutions in relation to Alfie and what can be done to help him."
Additional reporting by the Press Association