Comparing this to electromagnetism with source (in slightly different notation) with the Lagrangian
L=−14F2μν−12ξ(∂μAμ)2−JμAμ
and the corresponding time-ordered Feynman propagator for a photon
iΠμν((p)=−ip2+iε[gμν−(1−ξ)pμpνp2]
it can be seen that the parameter λ and ξ are related as ξ=1λ.
So the cas λ=1 corresponds to the Feynman-'t Hooft gauge with ξ=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.