Crop image keep layer name and not <Image>

How do I go about changing some default, so it keeps the original name?

I don't think you can.

You're running afoul of some Illustrator history and basic application intent here. Before it evolved into the multi-purpose tool it is now, Illustrator was built as a vector graphics creation and editing app. Collecting, editing (or cropping), and managing any number of raster images (let along a large number of them), were never express parts of its feature set.  Even just the simple image-crop feature is relatively new. So, your situation as described is bound to expose some shortcomings, and it has. If your images were linked when you placed them, the links panel may help.

Of course I haven't seen your file, and I don't know how or why it was assembled in the way it was, but it sounds to me like it might be better housed and managed in a page layout app.