There are so many things that your son can eat at one year. My son is 13 months old and he pretty much eats what we do for the most part. But you'd want to make sure you were eating healthfully. Here are a few ideas:
beans (kidney, pinto, black, garbonzo)
avocado
cut up fruit and veggies
rice dishes
soup
pasta
yogurt
muffins, bread, toast
scrambled eggs
graham crackers
tofu
cooked brocoli, cauliflower, edamames, peas, corn
waffles, pancakes, french toast
pieces of burger
cucumber, tomato,
peeled apple (golden delicious are pretty soft as far as apples go)
dried fruit like raisins, apricots, apple, banana, pineapple, mango, and papaya,
squash, yam
hot dog
olives
peanut butter and jelly sandwiches, cut up
If you like to have quick things on hand, check out the frozen food section and canned goods. Lots of quick veggie and fruit ideas.
Pretty much if it isn't hard like a raw carrot, you can serve them what you are eating, just in small chunks.
Have fun.