I know this is a weird concept but this is the only idea I have because I posted this a few years ago and wanted to update it...
------
Abnegation - No Gum - Too selfish to indulge in something. (If they had to have gum, it would probably be unflavored.)
Amity - Every Fruit Flavor (Natural) - They are always sweet and full of flavor. Plus in Amity they grow fruit, and their gum would be made of their crops.
Candor - Any flavor they want - The Candor doesn't care what others think and will chew whatever they like best. (It was hard to think of something, so if you have any other ideas, please comment.)
Dauntless - Cinnamon - (One of my favorite flavors) It's a strong and powerful flavor, kind of like the Dauntless. It's spicy which I feel like Dauntless would try to see how much they could chew at a time as a dare.
Erudite - Mint - Research shows that chewing mint flavored gum increases attention and cognitive performance.
Divergent - Bubble Gum - Apparently bubble gum is a mix of many flavors and Divergents are a mix of factions.
------
Tell me what is your favorite gum, faction and which gums represent factions to you!
Also, I applied to become a member of the staff, so let me know. :)