Yes. In Photoshop, the straightforward method is to enlarge the canvas, then place your name or logo in the new white area. Use Image > Canvas Size to add space below the photo, then add text or a graphic in that strip.

In Lightroom, this can be done through the Print module by creating a template with an Identity Plate containing your name, signature, or logo, then exporting/printing to JPEG. Standard Lightroom watermarking places the watermark inside the image, not in a separate border area.

One important note: a white strip below the image is mostly a presentation choice, not real protection. Unlike a watermark over the photo, it can usually be removed easily by cropping.