If the roommate is on the lease, the eviction would have to be by your cousin's landlord. If not - that is, if the roomate is a subtenant who has contracted only with your cousin, has lived there for less than a year, there is no lease, and rent is paid on a monthly basis - your cousin may at any time issue a thirty day written notice of termination of tenancy. If the same is true except the roommate has lived there for more than a year, a sixty day written notice must be given. If the roomate doesn't move out as required by the notice, your cousin would have to begin eviction proceedings.
If your cousin is alleging that the premises were used for an illegal purpose, he may be eligible to serve a three-day notice; but if he wishes to attempt that approach he would benefit from consulting a local lawyer who handles landlord-tenant matters.

