Are you able to set your name internally in a Nikon d60 DSLR?

Other cameras have the options to set the user for copyright info, and I have an older model Nikon d60 DSLR. It will be upgraded relatively soon, but until it is, am I able to set my name in the camera itself?

No.

If you copy your files over from your memory card using Adobe Bridge or Lightroom ("import"), it can automatically put your copyright info in the EXIF data as they're copied over, though.

Look on page 124 of your user manual.

You can insert an Image Comment up to 36 characters long

When preparing your RAW files for processing in Lightroom, you can add your copyright notice, keywords, contact information and modify the white balance using any custom white balance tool to produce a lighting "profile" like Xrite Colorchecker Passport.