As an avid user of the Engi ship, ion damage of x removes x bars of power from the hit system for 5x seconds. Hitting the system again will reset the timer. If you have a leveled up weapons man, you can ion shields and another system with the ion blast mark II. Generally, I go for oxygen. However, it is often best to just keep it on autofire at the shields. Also, I believe that ion damage does not allow you to add extra power to ioned systems. Therefore, a good tactic is to damage the shield room until there is only one level of shield and autofire ion weapons into shields. This will keep shields more or less permanently down if you do not let the timer run down. if it does, the shields will be fully repaired and will go back up.
This said, a good strategy for the boss is to have 2 ion weapons and at least 3 laser shots. In my case, I had the ion mark 2 and heavy ion, hull laser mark 1, and an anti-ship drone mark 1. Ions should be autofired into shields, and lasers should be fired wherever you want, probably weapons. Anything that damages personnel in the room they hit should be fine. If you have piercing weapons you don't necessarily need 3 laser shots, just 2. I recommend having a defense drone to deal with the missles. Those hurt. Level 4 shields are more or less required. Cloaking helps, but isn't needed.
When I beat it, I had level 7 weapons, level 4 engines, 4 shields, level 4 drones, level 1 cloaking, level 2 doors, level 2 sensors. In this fight, have all systems manned. If you get a 1-shield enemy with non-piercing weapons early game, just autofire ion into any system you want except oxygen. Using this method, you can level up all your crew in all specialties. This really helps late-game, especially the weapon cooldown and dodge.
If you're using a ship with 4 weapons slots, have 2 ion weapons with a total of 3 or more ion damage (take down 2 layers of shields), and at least 3 shots of laser. Another method would be at least a burst laser mark II and a beam. Go for pike or glaive because they have the longest beams and will most likely do more damage.
If you have ion mark II and a leveled up weapons, autofire on something other than shields. More often than not, you can keep the timer going. I haven't tried, but a level 2 weapons man should be able to have 3 systems ioned.
If you haven't leveled up weapons man yet, just focus on one system.