Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$?
Equivalenty, for which rings $R$ does there exist a cardinal $\kappa$ such that every module $M \in Mod_R$ is a direct sum of modules generated by $\leq \kappa$-many elements?
Faith  says that "$Mod_R$ has a basis" if it answers to Question 1.
- Kothe showed that if $R$ is an artinian principal ideal ring, then every $R$-module is a direct sum of cyclic modules (i.e. $R$ answers to my question with $\kappa = 1$).
The general question with $\kappa=1$ seems to be generally known as "Kothe's problem".
- Cohen and Kaplansky showed that the converse holds if $R$ is commutative. Warfield extended this to show that if $R$ is commutative, then $R$ answers to my question if and only if $R$ is an artinian principal ideal ring.
So the question is only interesting when $R$ is noncommutative.
- Nakayama constructed a (noncommutative) ring $R$ such that every module is a direct sum of cyclic modules, and yet $R$ is not an artinian principal ideal ring.
So Warfield's theorem apparently doesn't extend in the most straightforward way to the noncommutative setting.
- Faith and Walker showed that there exists a cardinal $\kappa$ such that every injective right $R$-module is a direct sum of $\leq \kappa$-generated modules if and only if $R$ is right Noetherian. Faith ( cf. also Griffiths Thm 2.2) then showed that if $R$ answers to Question 1, then $R$ is right Artinian.
So any ring answering to Question 1 is right Artinian, though not necessarily a principal ideal ring by Nakayama's result.
I happen to be interested in a special case:
Question 2: Same as Question 1, but assuming that $R$ is hereditary.
 Faith, "Big Decompositions of Modules" AMS Notices 18, Feb 1971 p. 400