This occurred to me when my phone number changed and the other information on my pets' tag was still correct. It is simple, cost effective, and weatherproof (so far, they had them a few months like this.) The best part is that it can be done with things you will probably already have in your home.
Step 1: Materials
Existing dog tag
Step 2: Adding Information
Use the permanent marker on the back of tag to write new information. I also did an arrow in the front to refer someone to the back. On my dog's current tag, the line i crossed off in the front I turned into an arrow, for a larger visible reference.
Step 3: Sealing It
Place tape over the written information and trim if needed. The tape should tear easily on the edges,taking the shape of the tag.
Step 4: Place on Pet and Enjoy!
Now you have peace of mind and a quick fix until you get a new tag!