While
total angular momentum is certainly conserved in any rotationally invariant system (read: Earth + space elevator + objects traveling upwards + atmosphere), the situation only for an object traveling to space in the elevator is different:
Angular momentum is given by
L = I * ω
where I is the moment of inertia and ω is the angular velocity. Let's assume that we put the internet (https://m.youtube.com/watch?v=iDbyYGrswtg) into the elevator and ship it off to space. Let's also say that the internet has a mass m and is of negligible diameter compared to Earth's radius. Clearly, its angular velocity ω will stay constant, assuming that the space elevator was built perfectly straight and perpendicular to Earth. However, the internet's moment of inertia I = m r^2 will not stay the same but increase quadratically with its radial position r and its angular moment will thus have to increase, as well. If you take the initial radius to be Earth's radius (~6,300km) and the final radius to be the geostationary orbit (~36,000km), the quadratic dependence will give you an enormous difference in angular momentum which will have to be accounted for by some torque that we apply to the internet.
It doesn't matter where this torque comes from--for instance we could imagine the internet to have some kind of jet propulsion engine, where the emitted gas's angular momentum would exactly match the internet's change in angular momentum, so that total angular momentum is conserved.
Moreover, it could actually be a small torque, provided that we move the internet at a small velocity in the perpendicular direction. (Recall that the change in angular momentum is given by the integral of the torque over time, so the smaller we want the torque to be, the slower we have to move the internet and the longer it will take the internet to reach its final position.)
Now what happens if we don't provide said torque? Recall that I assumed earlier that the angular velocity ω of Earth and the elevator (and thus the internet) be constant. If we don't provide the mentioned torque, moving the internet upward will increase the moment of inertia of the system Earth + space elevator + internet due to more mass being located farther away from the axis of rotation. By conservation of angular momentum L = I ω, the entire system's angular velocity then has to decrease, similar to a figure skater stretching out her arms while rotating in order to slow down. All this assumes, however, that the space elevator's cable is completely rigid in the lateral direction and doesn't move under the effective torque it will feel as the internet moves upward. In practice, however, this will not be the case unless further measures are taken which counter the effective torque on the cable and keep the cable perfectly perpendicular to Earth. (Only then Earth's rotation will actually slow down under the backreaction.) This seems somewhat difficult to achieve, though, and I think the better way would indeed be to equip the internet with a jet propulsion engine which provides the necessary angular momentum.
[EDIT] Made the part about backreaction on Earth and stability of the space elevator a bit clearer.