Lewis wrote, but never published, a very influential paper setting foundations for the multiplicative theory for Mackey functors. The paper is called "The Theory of Green functors" and Doug Ravenel's paper archive has a scanned copy. The date listed there is 1980, and there are numerous references to it in the literature. From the introduction:

Thus this project became, for the analogs of rings, a rough draft
equivalent of an undergraduate text on the basics of ring theory.

I strongly suspect that this is one of the influential works that Prof. May had in mind. These predate Tambara's work on Green functors that also possess a multiplicative transfer (now "Tambara functors").

On a more personal note, I once had correspondence with Prof. Lewis asking some questions - mostly trying to get some effective calculational methods in the homological algebra of Mackey functors. He wrote back a very kind response. I will attach some relevant snippets from his email:

The only thing I can suggest is an old idea of mine that I have have
intending to explore for years, but have never gotten around to
looking at. If there is any context in which this idea might be
useful, it is probably the one you are asking about [...]

Anyway, assign nonnegative integers to the subgroups of a finite group by
assigning 0 to the trivial subgroup, 1 to each of the cyclic subgroups
of prime order, and so forth so that the number assigned to each
subgroup is the length of the longest chain of subgroups starting with
that subgroup and ending with the trivial group. Filter any Mackey
functor M by looking at the kernel of restriction to the collection of
subgroups one notch below the whole group, the kernel of the
restriction to
all subgroups 2 notches below the top, and so forth.
This gives a natural filtration on the whole category of Mackey
functors [...] this is a very simple idea that lots of people may have
tried already and found useless. On the other hand, it may be
one that no one has tried because everyone assumed someone else had
looked at it or because no one could get a useful description [...]

Regardless of the specifics, I think the problem Lewis addresses here is one that is still very present in equivariant stable homotopy theory. Namely, when working with Mackey and Green functors, one often wants to induct on subgroups and try to filter calculations by simpler ones. In equivariant stable homotopy theory we often do so by using "isotropy separation" techniques, allowing us to attack one subgroup at a time (see, for example, Greenlees-May's paper "Some remarks on the structure of Mackey functors"). However, to my knowledge nobody has successfully figured out how to package this isotropy-separation information in a way that makes concept and calculation more approachable in equivariant homological algebra (eg. using such filtrations to produce spectral sequences with effective $E_2$-terms).

If it strikes you as worth pursuing, feel free. I'm not likely to
think more about it anytime soon.

Unfortunately, this is from February 2006, and he passed away about three months later; I regret not having the chance to discuss more.