What all can lydia do?

Deaton did pair them up before season 3A, "its not just someone to hold Du under, its someone who has a stong emotional connection to Du an emotional tether, Lydia Du go with Stiles" As for powers, Lydia is a lot like a beacon she can alert her pack if anyone is in danger, for now that is all that has been revealed, if she has any further powers they will be revealed in future seaons/ later episodes...

As a Banshee Lydia is connected to some Supernatural network of death, "the whispers of other banshees."

This causes her to hear voices in her head which often leads her to murder scenes. It also sometimes causes her to write oder draw messages from the voices subconsciously.

As a Banshee, she has the power of the Piercing Banshee Scream. The scream drowns out all other sounds so that she can focus on the voices.

Lydia can also use her scream as a offensive weapon to create a strong sound wave capable of throwing people and objects short distances. She can even amplify it to shatter a skull, which she has done.