This answer may be viewed as a bit odd and out of fashion by some, but one direction in which you could go--which intersects with commutative algebra, algebraic number theory, and discrete mathematics--is factorization theory.
Back when you were memorizing your multiplication tables in elementary school, the Fundamental Theorem of Arithmetic made that relatively easy in that there was only one way to break down a positive integer (greater than 1) into primes (which can be thought of as "building blocks", in a sense).
The generalization of this property that is discussed in the standard undergraduate abstract algebra sequence is, of course, the concept of the unique factorization domain (UFD), also called a factorial domain. The canonical example of an integral domain that does not enjoy unique factorization is the following example in $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5}\,\vert\,a,b\in \mathbb{Z}\}$:
$6=2\cdot3 = (1-\sqrt{-5})(1+\sqrt{-5})$.
One can use the standard norm function for rings of algebraic integers to show that 2,3, and $1\pm \sqrt{-5}$ are all irreducible in $\mathbb{Z}[\sqrt{-5}]$, and that the two factorizations of 6 above are distinct.
What's interesting, though, is that it turns out that while nonunit elements of $\mathbb{Z}[\sqrt{-5}]$ may not factor uniquely into irreducibles, any two factorizations of a fixed nonzero nonunit element have the same number of irreducible factors. Such a domain is called a half factorial domain (or HFD). In a short, two page paper in the Proceedings of the AMS, L. Carlitz characterized all HFDs amongst rings of algebraic integers via the (ideal) class group. Such rings have unique factorization precisely when the class group is trivial and are HFDs precisely when the class group is isomorphic to $\mathbb{Z}_2$. So, in the context of rings of algebraic integers, the size of the class group gives a measurement of "how far" we are from unique factorization.
Since Carlitz's seminal 1960 paper, there has been a lot of work done in factorization theory (and yes, this was my area of research while I was in academia, so I'm not exactly unbiased). Currently, the only book on the topic is Geroldinger and Halter-Koch's Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Make no mistake, this book is an extremely dense read, but it's very thorough and has an excellent bibliography.
Other sources (some of which are a bit more accessible to the newcomer) include:
If I might make a suggestion.
I don't know the man, but the heartfelt words you expressed above say it all. If it were me, I would love to hear that from someone I've helped.
Perhaps finding a wonderful inspirational quote and then personalizing that with both your heartfelt sentiments would be worth gold to someone as they feel like they continue to pass on their good deeds!
I would even go as far as making it an inscription on a plaque of some sort. I would cherish such as thing forever and would display in a place of honor if I received that.
Both of you young people give us all hope!
"What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises." - G. C. Lichtenberg
Regards -A
Best Answer
The said legacy is mixed. The volumes dealing with algebra and particularly Lie theory are thought to be excellent. The volume Theory of Sets is not merely out of date but contains serious errors analyzed by a professional logician named Adrian Mathias (see my answer here).