You’ll get three options in the upper-left of the window that’ll appear—“Replace Text,” “Add Text,” and “Format.” If you pick that first option, you can search for a common element in your selected files and replace that with different text.
Choosing the “Add Text” option means that you can insert text either before or after your filenames.
Finally, “Format” will let you append the date or a number (e.g., “File 1,” “File 2,” etc.) to your filenames. You’ll pick what number you want to start with on that screen, and again, you can select whether to put the new data before or after the name.
And whatever renaming method you choose, be sure to check out the lower-left corner of the window to get an example of how your renamed files will look before you apply your changes.
Neat! Obviously, this isn’t full-featured enough to take the place of some third-party programs (A Better Finder Rename is popular, for example), but for most users, this built-in ability will be enough to meet their batch-renaming needs.
This tip was suggested by the very clever Bryan Mahler. Thanks, Bryan!