  Let $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus
\mathfrak{g}_{\overline{1}}$ be a Lie superalgebra over an algebraically closed
field, $k$, of characteristic 0. An endotrivial $\mathfrak{g}$-module, $M$, is
a $\mathfrak{g}$-supermodule such that $\operatorname{Hom}_k(M,M) \cong k_{ev}
\oplus P$ as $\mathfrak{g}$-supermodules, where $k_{ev}$ is the trivial module
concentrated in degree $\overline{0}$ and $P$ is a projective
$\mathfrak{g}$-supermodule. In the stable module category, these modules form a
group under the operation of the tensor product. We show that for an
endotrivial module $M$, the syzygies $\Omega^n(M)$ are also endotrivial, and
for certain Lie superalgebras of particular interest, we show that
$\Omega^1(k_{ev})$ and the parity change functor actually generate the group of
endotrivials. Additionally, for a broader class of Lie superalgebras, for a
fixed $n$, we show that there are finitely many endotrivial modules of
dimension $n$.
