There are many apps that target particularly annoying sounds. Not surprisingly, most have "annoying" in the titles. Use these to target environmental sounds that are disturbing to an individual. Have the student control the volume when experiencing them.
The school environment has a range of troubling noises. This app has them: fluorescent lights, coughs, blowing noses and clearing throats. Imagine the school cafeteria. It can be overflowing with upsetting sounds. Look for slurping and smacking lip sounds under the "human" category to help kids desensitize.
Here are some other annoying sound apps (all free). Both have a quick on-off switches and a repeat button. Volume needs to be controlled by the iPad volume button. Annoying Sounds! has twelve sounds, including hammering, in-sink-erator and garbage truck. Annoying Noises comes with 15 sounds including dripping, buzzing fly and a drone.
If you feel you need more control for your Avoider, try Sounds Annoying! You can change the pitch, volume and filter using slider buttons. Did you not find the noise that you need to target? It has the ability to customize your sound if you buy the in app purchase option (99 cents). If there is a timer beep, a certain tick of a clock or a voice that may be grating to a particular individual, record it and use this for controlled exposure.