Export every mesh into seperate .obj-file with according name?

I have a lot (400+) of objects (meshes), I need to get into Unity, each as a seperate .obj-file with the mesh-name as filename.

All the exporters I tried only export it as a single lump of objects with a generic name.
Any ideas how I can do this without spending hours exporting and naming them one by one?

Script, it takes first selected object name and call exporter and export file with object name. Then next selected object…loop.

This is exactly what I do, I have over 400 objects in one blend, in many different scenes and I export them all to separate .fbx files. The trick is that you need to add all of your meshes to their own object groups. Then you can just use the Batch mod: Group to get every group in your blend as separate .fbx file.