Perhaps the most important function is flavor. But because brown sugar contains traces of molasses, it is slightly acidic. Combined with baking soda, then, it leavens the cookie. The resulting cookie is dark, chewy and soft. A good cookie, perhaps, but not a typical chocolate chip cookie.
Eggs provide protein and flavor.
The protein, which sets quickly at 375 degrees, contributes to the preferred rounded shape of the cookie. The flavor comes from egg yolks. Some cooks complain that cold eggs make the butter-sugar mixture look curdled. It doesn't matter to the final outcome of the cookie. Unlike cakes, cookies don't rely on room-temperature eggs to be successful.