Discrete Log with regard to a random base Suppose DL is hard in $G=\langle g \rangle$. For a uniformly random group element $r \in G$ (suppose $r=g^a$), is it hard to find $s$ given $r^s$ and $r$? Does the computational assumption have a well established name?

I can show it at least as hard as DDH, because if $\mathcal{A}$ can find $x$ given $g^y$ and $g^{xy}$, she can distinguish $(g^x,g^y,g^{xy})$ from $(g^x,g^y,z)$ where $z$ is a random group element.
You are asking if it is hard to find $s$ given $r^s$, but you do not mention if $r$ is also given.

If $r$ is not given, then the problem is: given some value $y$ find $r$ and $s$ such that $r^s=y$. This is an easy problem - just outputs $r=y$ and $s=1$.

If $r$ is given, then the problem is: given $(r,y)$ find $s$ such that $r^s=y$, where $r,s$ are randomly chosen. This is a form of the discrete log problem where the base is random. This makes it ``harder'' in the sense that the adversary cannot preprocess. I don't know if it has anything more established than this.