If instead one considers mapping into the stabilization of the spheres, hence into (some suspension of) the sphere spectrum, then one speaks of stable cohomotopy. In other words, the generalized (Eilenberg-Steenrod) cohomology theory which is represented by the sphere spectrum is stable cohomotopy.
In this vein, regarding terminology: the concept of cohomology (as discussed there) in the very general sense of non-abelian cohomology, is about homotopy classes of maps into any object (in some (∞,1)-topos). In this way, general non-abelian cohomology is sort of dual to homotopy, and hence might generally be called co-homotopy. This is the statement of Eckmann-Hilton duality. The duality between homotopy (groups) and co-homotopy proper may then be thought of as being the special case of this where is taken to be a sphere.
Wikipedia, Cohomotopy group