...I would prefer to try to scrape it all off...
Easy to say, but most of us aren't quite as ambitious as you, Glenn
. To my experience, the so called stucco is just mortar or sand mix, probably troweled over the wall in lieu of badly needed tuck-pointing, which in turn is due to moisture wicking through the brick and leaving cakes calcimine. I'm guessing that where the mortar is pulling away there is white powdery stuff under it. That's the calcimine.
What Glenn says
is the right way to fix it, but it's not a task for the faint hearted. Unless you plan on finishing the basement for extra living space, you're best off to follow guyod's advice and just patch what's coming loose. One thing he didn't mention though, is that you should clean it of any loose dust before patching it, but I guess that's just common sense. The best way to clean it is by blasting it with the garden hose, if that's an option. If not, get a good quality dust mask, the kind with 2 rubber bands, and go at it with a corn broom or a stiff brush. Then wet it just before mashing new mortar into it.
You'll probably have to do it again in 5 or 8 years, when more areas have come loose. One way to help prevent this from happening is to make sure that rain water drains away from the house. Where the exterior grade is not pitched away from the house is where you are most likely to get moisture seeping through the brick foundation.