I'm pretty sure boxed pudding mix is mostly cornstarch, sugar, maybe some powdered milk or powdered eggs, flavoring, and then dyes and preservatives. If you just dumped a box of pudding mix into a basic lean dough (just flour, water, salt, and yeast), you'd end up with something close to a typical enriched dough (lean dough plus stuff to make it sweeter, more tender, etc). Obviously the sugar and flavorings are gonna sweeten things, and the cornstarch might have a tangzhong-like effect where it traps water, leading to a softer, moister, more tender finished product. It'll also probably interfere with gluten formation, which will also lead to a softer, more tender dough.

To figure out what it could replace, let's consider what's in a "normal" cinnamon roll dough first. Commonly a typical cinnamon roll dough is basically brioche dough, so a lean dough enriched with eggs, a touch of sugar, and a healthy amount of butter. Egg yolks, sugar, and butter all interfere with gluten formation and lead to a softer dough, while egg white might lend a bit of structure, but realistically is mostly just contributing water.

So the most obvious thing that's being replaced is the sugar. If the pudding mix contains some sort of powdered dairy product, that might lend some dairy flavor, but you'd still need some sort of fat. If the pudding mix contains powdered egg, that might lend some egg flavor, but powdered egg has less fat than fresh, so again you may need to supplement there as well. If the pudding mix contains cornstarch, I'd consider lessening the amount of flour in the dough to make sure it's still at the right hydration level.

Note: I've never done any of this myself, so this whole thing is basically just an educated guess 😅