Sometimes one requires just that the inclusions are null-homotopic map?s. This might be called semi-locally contractible.
One could also consider a basis of open sets such that the opens have (just) trivial homotopy groups, but this does not seem to crop up in practice.
This is discussed at locally ∞-connected (∞,1)-site.
If one considers fundamental ∞-groupoids, the inclusion being null-homotopic is equivalent to the induced (∞,1)-functor being naturally isomorphic to the trivial functor sending everything to a single point.
David Roberts: The following may be straightforwardly obvious, but I have couched it as a conjecture, because I haven’t seen it in print.
If the space is semi-locally contractible then every locally constant -stack on the site of open sets of is locally trivial.
See also locally ∞-connected (∞,1)-topos. There a converse to this conjecture is stated:
Let be a site coming from a coverage such that constant (∞,1)-presheaves satisfy descent over objects of with respect to the generating covering families. Then the (∞,1)-category of (∞,1)-sheaves is a locally ∞-connected (∞,1)-topos.