I've seen it spelled and punctuated four or five different ways. What's the textbook way, and why? Was the species discovered by someone named Hermann, was the first one named Herman ...
It is Hermanni. The tortoise was named for a man who's last name was Hermann. So if you say Hermann's, you're saying that the tortoise belongs to Hermann. If you say Hermanns' you're saying that there are more than one Hermann tortoise and the apostrophe at the end means something belonging to those hermanns tortoises. Like, "That is all my Hermanns' dinners." There should be very few instances where you would use the apostrophe at the end of the name.
We say leopard babcock tortoise, not leopard babcock's, babcocks tortoise.
We say sulcata tortoise, not sulcatas tortoise.
So I think we should also say Hermann tortoise, not Hermanns, Hermann's.
The -i suffix in Latin means "possession of a male," and the -ae suffix means "possession of a female." Thus, the scientific name Testudo hermanni literally means "Hermann's tortoise." The common name is often shortened to just Hermann tortoise, but unless you're referring to multiple individuals, you wouldn't write "Hermanns," because that's a plural.