Final answer:
The axillary nerve, originating from the brachial plexus, gives rise to the upper lateral cutaneous nerve of the arm.
Step-by-step explanation:
The nerve which gives off the upper lateral cutaneous nerve of the arm is the axillary nerve. This nerve is a systemic nerve that arises from the brachial plexus, which itself is made up of spinal nerves C4 through T1. The axillary nerve branches out to go to the armpit region, with one of its branches being the upper lateral cutaneous nerve of the arm, and is instrumental in the sensory and motor functions of the shoulder area.