When renting in the private sector there are two things to consider (1) what is the maximum benefit available (this does not always cover the rent in full) and (2) is the maximum entitlement reduced by anything - i.e. income/capital.
(1) what the maximum benefit available is, will depend on a number of factors - your son's age, where he lives, benefit cap (if applicable), whether he is in a Universal Credit area, etc. Assuming he is not in a UC area, he can find out the rate of 'Local Housing Allowance', which is the maximum benefit available, on direct.gov follow this link:-
If he is under 35 years, he may well be subject to the 'shared room rate' even if he is not sharing, this applies unless he receives high rate care DLA.
(2) if he only receives income based ESA/DLA and has capital of less than £6k then he should receive the full 'allowance' whatever that may be in his area.
If there is a shortfall between the allowance and his actual rental liability, he can apply to the local authority for a 'Discretionary Housing Payment'. However he will have to make a good case as to what exceptional reasons he has for needing the higher rent, as the schemes are notoriously hit and miss.
Hope this helps.