Well she asked if it could be called something else beginning with c. Courgette begins with c and is another name for a zucchini. Just trying to be helpful.
No, I would have remembered if it was courgette.
I think I found the answer though. I also posted a thread on a food forum I go to and someone came back saying it looked like a coosa/kusa. They posted some pictures that look like mine except longer, coosa also looks like the word that was on the sign at the farmer's market. Thanks everyone! I can't wait to try it!

ETA: I just also saw that they can be called Italian squash or cucuzza. The picture earlier didn't show one with a stem that looks like mine, so I didn't think it was the same thing, but apparently it is. And, the person also said that if it gets longer it becomes a marrow. I guess the small ones can be cooked like a zucchini, but the long "marrow" ones should be cooked like a squash.