As an update, we bought the most recent set of D&D books, version 4, so we could play this game. Big mistake! It's so much more complicated than the original game I learned. So, in the spirit of "everything was so much better when I was a kid," I decided to find the old books. There are a couple of options.
[Update 1/9/09: OK. Once we ignord most of the melee rules, like cover, advantage, lying prone, etc and then made cards for the once-a-day/encounter/at-will powers for the dad to have at his fingertips (and to help advise the kids which they could use at that time), the 4th edition is working out pretty well. That every class has various powers they can use and all of the skills that are different for everyone makes it much harder for the kids to use their character effectively--but the skills may not be necessary if you are designing the campaign to be a bit easy on them. Again, this is something the Daddy DM may try to help the kids to remember. In the end, we decided to use the 4th edition after all.]
The easiest option seems to be The Dungeons and Dragons Basic Rules set. Finding the complete set is difficult and more expensive than necessary. You can easily find the basic rules book (approximately 64 pages) and an additional book that is an adventure to follow (easiest option if you're new to D&D). The very first adventure is called The Keep on the Borderlands. You can also find the sets on Amazon and eBay.
The most available option is called Advanced Dungeons & Dragons. It's important to remember that Advanced Dungeons & Dragons requires three books: The Dungeon Master's Guide, The Player's Handbook, and The Monster Manual. The books were all published in the early 80s. One thing to watch is that there is also a 2nd Revision (usually visible on the cover if not in the product description). I don't remember how much of a difference there is, but I stayed with the 1st Revision (generally not specifically marked on the cover, but sometimes in the description).
All of those books were published by the original game publisher TSR. The third revision was published by a new company and a lot of rules were changed. There are other options by the new publishers for more simplified games. They too have a basic set. There is also a version of the game that is simplified and goes along with purchased miniature figures.
It was quite a trip down memory lane for me and my husband.