Absolutely. You can take your one-year-old with you on any ride for which there is no height requirement. This includes many of the most popular rides at Walt Disney World.
If you look at this page of travel tips for people with small children found
HERE, you will see a button on the right hand side of the page that says "Height Requirements." Clicking on this link will bring you to handy pdf that you can download. The pdf lists the attractions in all four theme parks and both water parks that have height requirements. If an attraction is not listed on this document it means that there is no height requirement and any age child can enjoy the ride. For example, your son can join you on Dumbo, Small World, the Many Adventures of Winnie the Pooh, Cinderella's Carousel, Peter Pan's Flight, and many others. My daughters also loved the Country Bear Jamboree at that age, particularly the talking moose on the wall.
Also remember that they are activities other than rides that your son may enjoy. When my daughters were one, their favorite activity was meeting the characters. We spent a lot of time in Toontown Hall of Fame meeting characters. They loved cuddling with Mickey and his pals. Your son might also like meeting characters at a character meal. Favorites of ours are the buffets at the Crystal Palace in the Magic Kingdom and at Chef Mickey's in the Contemporary.
Your son will likely also love spending time at your resort pool with you. In fact, we were often having such a good time splashing in the water, that we almost forgot there were theme parks to visit.