If I only want to use part of a can of beans, what are my options with the rest?
As rfusca says, they'll be fine for a few days in the fridge; HOWEVER, if you want to transform them into something more delicious and less perishable to go with eggs,  you can cook them into black bean fritters. 

First, cook the beans with garlic, minced onion, and chile powder to soften them.  Mash or puree, and reduce until thick, then add salt. Done right, this can be pan-fried into something delicious to go with eggs.  Probably one of my favorite breakfast items from a summer in Guatemala.

Depending on how much water you remove from the fritter base, it should be fine in the fridge for 3 days to a week. It can also be shaped into patties, stacked between layers of wax paper or baking parchment, and frozen indefinitely.