Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?

I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable categories. Here are two references:

• Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd, Reflexivity and dualizability in categorified linear algebra, Theory and Applications of Categories, Vol. 30, 2015, No. 23, pp 808-835. [link] See Lemma 2.7.

• Julia Ramos González, On the tensor product of large categories, Thesis, University of Antwerp 2017. [link] See section 2.4.

I say “Deligne-Kelly tensor product” for this. Technically Deligne tensor product is for certain abelian categories and right exact functors, and Kelly is for finitely cocomplete categories and right exact functors, while here you want locally presentable categories and cocontinuous functors. But they’re all similar enough, see Section 3.1-3.2 of Ben-Zvi-Brochier-Jordan for a quick summary and some references.

(Aside: If you want understand the technical differences between Deligne and Kelly's versions the must-read paper is Lopez Franco.)