/data/system/dropbox is not part of the cloud storage provider "Dropbox". It is part of the Android OS DropBoxManager (not related at all to "Dropbox", just happen to have the same name).
This puts data from application crashes and kernel logs and such in this log directory. I know the market uses this directory (See #4) as well, probably to report application crashes.
You could probably remove the data in this directory, but DO NOT delete the directory and you may see logcat errors from parts of the OS/Apps looking for specific files in this directory.
Edit: I deleted everything from this folder and rebooted my device afterwards, I had no issues with any applications, and it actually freed up some space on the device.