What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?

One can call it pure object. 


In the context of tstructures, it's not unreasonable to call this just a "sheaf". That is, in the natural tstructure on the derived category, the core is exactly the full subcategory of these things. In the perverse tstructure it is, by definition, the category of perverse sheaves. If one can say that a complex with vanishing positive and negative perverse cohomology is a "perverse sheaf" surely one can also say that a complex satisfying the vanishing of positive and negative cohomology is just a sheaf. 

