Asssuming England/Wales. I assume your original tenancy was an Assured Shorthold Tenancy, this will have rolled over to a Statutory Periodic Tenancy.
New contract? You don’t really need one. If son has inherited that would be implied in normal contract law anyway. A note or addendum to the original could be handy if you need the contract for other purposes though. (do you need to show contract for housing benefit claims, or to prove residency for immigration type queries?)
The trouble with signing a new contract is it ties you and the new landlord in a bit tighter than your current position in the first few months if you sign up for an assured shorthold tenancy, which is usual. Your potential problem here is that if you sign up for, say, 6 month AST and then find you don’t get on with new landlord, you are legally tied in for the period and might have to wriggle a bit to move on quickly without additional cost. From the landlord point of view, if they sign you up on AST, you have more protection in the initial period should they want you out.
To put you back into an equal position before the death, if a new contract is to be signed would be to go straight for a periodic tenancy, with an initial period of one month.
Edit: I’m not a lawyer, but a reasonably experienced landlord.