A category is (sometimes) called pointed if it has a zero object. Every such category (at least if locally small) may be interpreted as enriched over the category of pointed sets using the smash product as tensor, (but not necessarily the other way around).
The terminology certainly comes from the fact that every category of pointed objects has a zero object: the point.
In the homotopical context of a pointed category of fibrant objects many of the familiar constructions from homotopy theory in Top have analogs.
Notice that this notion of pointed is not related to usage such as in well-pointed topos. Nor is it to be confused with a pointed object in Cat, which is just a category equipped with a chosen object (which need not be initial or terminal).
A slight weakening of the concept of a pointed category is that of a quasi-pointed category.