I have a Sony W690 PointAndShoot camera, and I'd like to know if it's possible to add my name to each photo that I take, similar to adding the date on our pics. I have seen pics that have the photographer's name.
The related questions that Esa Paulasto gave links to them are great to understand what is a watermark and helps you understand the cons and pros.
If you want to know if it is possible to add a watermark automatically via your camera like the date that was added, then NO. Moreover, even if it was possible it would probably be a bad idea
It might cover something important in your shot and will ruin it.
You can forget it "ON" and use it on your best friend wedding. Wedding/family photos with watermark looks odd to me and definitely unnecessary.
I took a quick look thru the Sony W690 user manual, and it looks like there's no option for freely chosen text (your name) being added on the photos.
But you can do it with a photo editor, only it happens in computer, not in-camera. The word "watermark" sounds a bit too fancy, when at its simplest form it could be just plain text drawn over the photo, preferably in a corner or such a spot where it does not jump on your face when looking at the photo. See "How do I position a watermark?"
You can also add your name and other info inside the photo so that it does not become visible on the photo but stays inside the data of the file. That info is called metadata, and also with word EXIF. You can use a photoeditor to write into metadata, or with some cameras you can put some info automatically into each and every photo you take. See "Adding data to an EXIF file". Though that info stays inside the file and is not showing on the photo if and when you print it on paper.
I'm not sure if your camera supports it or not, but another alternative to visibly watermarking an image is to embed your name and a copyright message in the EXIF data of the image. I know my camera automatically does this for me, but I also have a high end DSLR.
I'm not sure if it is possible for your camera, but if it is, it would put information in the photo that informs people about who took it without disturbing the image itself. (You can also normally get the date from the EXIF info as well.)
You could also then use just about any watermarking program to take the copyright information of the image and place it over the image if you wanted to watermark it.
It is possible on certain Canon point and shoot cameras which can have their features extended via the Canon Hack Development Kit (CHDK).
This example outlines some ways in which a watermark might be useful at the time of recording and shows how a perceptual coding / hidden watermark being added to captured images in near real time. With a bit of work this technique could be used to overlay an image of your choosing instead.
As to if it's wise to watermark your only copy of an image (especially if the camera doesn't detect orientation correctly) - that's a whole other ballgame and the CHDK wiki a user asks about overlaying date/time using CHDK and is met with a brief and dismissive response.