In addition to mining dilithium, you can also get it by defeating certain types of hostiles. Pakled Thieves and Sona Pirates will both reward you with dilithium when you defeat them.
Pakled Thieves can be found in the following system:
Sana Pirates can be found in:
At the moment, my fastest mining ship/crew can extract 480 units of dilithium per hour. Quick, 10-minute (plus 5 minutes for travel and repair) raids in Coridan, Bellas, and Elona let me get around 550, 650, and 850 units respectively. The NPCs just keep coming, so you don't run out of targets, but you do take hull damage. I try to keep repair time to a minimum, so I keep the raids short. It's a lot faster than mining, but it doesn't have the same "set it and forget it" advantage. Hunting down and blowing up hostiles as fast as you can takes a little more effort, but it's worth it.