Is there a standard name for the type of singularities a codimension-$1$ subvariety of a smooth algebraic variety has when it looks locally (possibly analytically) like an arrangement of hyperplanes? «Normal crossings» refers to the special situation in which locally it looks like a subarrangement of the Boolean arrangement, that is, some coordinate hyperplanes.
This paper uses the term "arrangement of smooth, complex algebraic hypersurfaces", or simply, "arrangement of smooth hypersurfaces". To quote: "Our goal here is to further generalize these results to a much wider class of arrangements of hypersurfaces, by which we mean a collection of smooth, irreducible, codimension 1 subvarieties which are embedded in a smooth, connected, complex projective algebraic variety, and which intersect locally like hyperplanes."
In another paper on the subject, Clément Dupont simply uses the term "hypersurface arrangement". To quote: "We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties."