I threw the following together before I saw Jim's response., but here goes:
Angular momentum is conserved - even within a black hole, just as mass is conserved within a black hole. It's one of those properties that tends to remain constant in a universe outside unusual environments.
Black holes, of course are fairly unusual environments, so the physics of a rotating black hole are a lot more complex than those of a stationary one.
Nevertheless, the angular momentum of a black hole is still (kind of) the sum of the angular momenta of the contributing masses.
But, it depends a bit on your frame of reference.
A body falling onto the BH might be spinning on its axis, and so has some angular momentum, but it might also be orbiting the BH in one specific plane, so it has some angular momentum from that motion as well. The amount of angular momentum added the BH when the object falls in depends on the frame of reference chosen to measure the angular momentum.
In a classical case, and assuming the body is a long distance from the singularity, so that the forces on one side of the body are not significantly different from the forces on the other side, the body falls toward the singularity.
It is rare for the trajectory of the falling mass to be directed perfectly at the singularity, so the body falls in the general direction of the black hole with some momentum along a particular trajectory.
Even if the falling body is falling directly toward the singularity, there is likely to be some extraneous matter orbiting the BH, that will deflect the body away from its targetted trajectory.
As the gravitational forces from the BH act on the body, they accelerate it in a direction directly in line with the singularity. This gravitational force transfers some Angular Momentum from the BH to the falling body (just as a man-made satellite can slingshot around Jupiter gaining angular momentum from the interaction with Jupiter's gravitational field).
Depending on the mass of the BH, the orbital radius of the falling body becomes closer and closer to the BH. In order to conserve angular momentum, the orbital speed accelerates. Depending on the mass of the BH, the forces involved in this tight orbit might tear the falling body apart through mechanical forces.
If the BH has a very large mass, the gravitational gradient at the event horizon is relatively shallow, and the falling body moves into an unstable decaying orbit around the BH. Eventually, the falling body moves beyond the event horizon, and we can know no more about its trajectory, though we estimate that it continues in the decaying orbit, until tidal forces rip it apart, and the angular momentum is transferred back to the BH.
At least, that's how I think it works. Happy to be contradicted.