Yes, with DropIt you can also rename and move your files with a single process.
You only need to correctly configure an association with Rename action.

This is an example of association that adds the image Date Taken at the beginning of my photos:

Name: Move and Rename my Photos

Rules: *.jpg

Action: Rename

Destination: C:\Example\Of\Destination\%DateTaken% %FileName%.%FileExt%

For more info about abbreviations (the special words between % percentage symbol) you can read:
How to customize destinations