My birds like to chew on my wandering jews, bamboo and hen and chicks.
Hen and chicks are super good, as they don't really need a lot of soil, and they basically constantly repopulate themselves, and basically thrive without much water. Every 4-5 years or so, the main hen will sprout a flower and die, but by then, you should have more than you know what to do with. Just make sure not to get the cobweb/spiderweb variety. I'm not sure if it's toxic, but they have this thin sort of silky spiderweb coating the plant, and I'm very hesitant to let my birds at the chicks I had and got rid of them.
Wandering jews are also very pretty. They're a hanging plant and easy to propagate after being broken, and my birds seem to be attracted to the colors. I have both the big-leafed purple and smaller leafed purple-and green variety. The smaller one with the purple-green leaves seem to really catch my birds' eyes, as the leaves are shimmery. The flowers they put out are also really pretty. Guzma had a grand old time pecking at the pink ones.
Bamboo is a bit hit and miss with my birds. My tiel enjoys pecking at the leaves mine put out, but my budgie is sort of iffy with it. She'll perch on it, but that's about it.
Aloe vera is also very good for birds, but not as an in-cage sort of plant. The outside of the plant can give birds the runs, but the goopy insides are very good for birds. I always mix it in their water.
My birds also love to chew on my winter pear tree's leaves, but they get way too big for an inside-plant.