  We prove that ideal sheaves of lines in a Fano threefold $X$ of Picard rank
one and index two are stable objects in the Kuznetsov component
$\mathsf{Ku}(X)$, with respect to the stability conditions constructed by
Bayer, Lahoz, Macr\`i and Stellari, giving a modular description to the Hilbert
scheme of lines in $X$. When $X$ is a cubic threefold, we show that the Serre
functor of $\mathsf{Ku}(X)$ preserves these stability conditions. As an
application, we obtain the smoothness of non-empty moduli spaces of stable
objects in $\mathsf{Ku}(X)$. When $X$ is a quartic double solid, we describe a
connected component of the stability manifold parametrizing stability
conditions on $\mathsf{Ku}(X)$.
