I apologize for the responses of my friends and colleagues above. The are well meaning and think they are witty, but few demonstrate the true understanding of the problem.
I comiserate because I too adore the instrument and have had cause over the years to nurse more than one kazoo back to - well if not full health , at least to serviceability.
My experience with the beast is that when replacing the diaphragm, you need to find the correct balance of looseness, razz and tension. The balance that will enable your hum to grow and express itself. This is a process of trial and (often much) error, resulting in the sort of quest you describe.
Take some normal kitchen grade wax paper - about 5 cm square. Crumple it to oblivion and the straighten it out again in a more or less smooth sheet. Take the sheet and stuff it into the sound hole of the kazoo so that it is not taught but sinks below the rim by about the distance of a thumbprint. Then tighten up the collar and hum. If it does not resonate enough, slacken the collar and press it down a tiny bit more. If it doesn't resonate at all it might be too loose.
As noted in the many messages above there are all sort of materials which can be used. There are indeed pre-made kazoo diaphragms with cardboard collars. Wax paper is my standard but I have used and had pleasing buzzes with saran wrap and even bond paper (the latter only when properly crumpled first)
Experiment. Find what works for you. The kazoo is a great instrument. It is perfect for solos in Big Band piece's like Benny Goodman's "Boodle-am Shake" and as a stand-in for Leon Redbone "Throat Tromnet" growls. Don't be swayed by peers who do not understand or love it as you do.