Comparing this to electromagnetism with source (in slightly different notation) with the Lagrangian

\[\mathcal L = -\frac{1}{4}F^2_{\mu\nu} - \frac{1}{2\xi}(\partial_{\mu}A_{\mu})^2 -J_{\mu}A_{\mu}\]

and the corresponding time-ordered Feynman propagator for a photon

\[i \Pi^{\mu\nu}((p) = \frac{-i}{p^2 + i\varepsilon}\left [ g^{\mu\nu} - (1-\xi)\frac{p^{\mu}p^{\nu}}{p^2}\right]\]

it can be seen that the parameter $\lambda$ and $\xi$ are related as $\xi = \frac{1}{\lambda}$.

So the cas $\lambda = 1$ corresponds to the Feynman-'t Hooft gauge with $\xi = 1$. The only advantage of this is that it makes the propagator consisting of only one term, but there is as said in the comments no physical meaning of this.