Well, with Pythagoras you could talk about more than just the Pythagorean theorem. You could talk about the Pythagoreans and how their philosophy connected to mathematics. Still, I wouldn't personally pick Pythagoras.
With Euclid, you could discuss his Elements as well as the revolutionary idea of axiomatized argument. You could touch on the explosion of rigor starting around the 19th century and continuing into the present day [Weierstrass et al founding analysis; the foundational crisis and its resolutions, etc.].
With Archimedes you don't really have to go through physics. You could instead discuss his early form of calculus and lead into a discussion of calculus through the ages (which is a large topic).
Another thread you might follow is the development of non-Euclidean geometry. In particular, you could go over the rather substantial development of spherical trigonometry in the ancient world and trace the development of hyperbolic geometry through the 19th century. It wouldn't be difficult to connect this all to Einsteins' theories of relativity and Riemannian geometry.
You could also pick an ancient civilization and discuss their contributions to math. Ancient Indian mathematics tends to be interesting. You'd want to focus it more, somehow; maybe picking a time period or a type of thought.
Diophantus might be nice to study. He connects with lots of topics like Hilbert's 10th problem, Fermat's investigations into number theory (and many others), and Fermat's Last Theorem.
If it's a shorter paper, I've always found the exile of Jewish mathematicians from Nazi Germany interesting. There's a wonderful quote from Hilbert where he was asked how math was in Gottingen since the Jewish presence had been cleansed, and he responded with something like "Mathematics in Gottingen? Why there is none to speak of." I remember Noether was forced out, and certainly others were as well.
That reminds me, you could discuss women in mathematics as well. There aren't many, historically, but you could certainly discuss Sophie Germaine and Emmy Noether.
You could make a rather long paper on the development of calculus by Newton and Leibnitz, along with a discussion of the ensuing priority dispute. Bringing in Robinson's non-standard analysis, and maybe even standard analysis, might not be amiss.
And then you could do a paper on the "super geniuses" of math: depending on who you ask, Euler, Ramanujan, Archimedes, Gauss, and Hilbert, say. Certainly some of their problems feed into each other, though organizing the paper would probably be the hard part.
Best of luck. I hope something here is useful.