What are effective strategies for countering Mind Control

Some subsystems have a small "head" icon above their power levels. These systems are:

• Piloting
• Shields
• Weapons
• Engines

By staffing these systems with a crew member, you increase the abilities of that system. When you send a crew member into an unstaffed room, you can see them head straight for the console and start working there.

From my observations:

• Staffing the Cockpit and the Engine Room increases your chance to dodge.
• The Engine Room also has the added benefit of reducing FTL charge time when staffed.
  • Please note that staffing the Cockpit is required for almost everything; the engines won't recharge even if someone is manning the Engine Room. Your dodge chance also goes to 0% if no one is trying to pilot the ship; it's an inert hunk of metal.
• Staffing the Shield System results in shields recharging faster.
• Staffing the Weapons System results in your weapons charging faster.
• Staffing the Door Control System results in your doors upgrading a level.
• Staffing the Sensor Control System results in your sensors also upgrading a level.

The amount of the benefit you receive is based on the crew member's system skill level, which you can see by going to Ship --> Crew, and hovering over the crew member. They will gain experience by working on the system, increasing the bonus they provide to the relevant system. You can occasionally see your crew gain experience by seeing yellow icons appear above them. These icons will match the system they are currently working on.

Note that it doesn't seem like crew gain experience when the ship is not in danger, so staying in a system doing nothing is unlikely to help your crew gain experience.
The exception to this, as @Halfwarr noted, is asteroid fields. Since your ship is continually in danger and the asteroid damage can be mitigated, this is probably an excellent way to gain skill. This is no longer valid with the full release.

Also, there is only a single console per system, no multiple crew members do not provide any bonus beyond faster repair speed in the event the system becomes damaged. This is easily testable by sending another crew member to the cockpit; the first one will provide a bonus to dodging when he reaches the console. The second will not increase your ability to dodge.

I have not unlocked all ships, however, so it may be possible that some of the later ships have multiple consoles for some or all systems, in which case, multiple crew members would provide a greater bonus.

When boarding, I tend to aim for the Oxygen subsystem room - it's typically small, and critical enough that I will quickly attract attention. If they don't respond, then I've pretty much sealed their death quickly.

Meanwhile, I'll target whatever critical system I choose (usually weapons, shields, drones, or engines) with my laser weaponry. Once I've hurt one or more of their crewmembers, I'll target the medbay with missiles or bombs. That way, I can finish off heavily wounded crewmembers while disabling their ability to heal. Fire bombs and subsystem bombs are the best for this sort of attack.

I do believe that firing on the rooms where your boarders are will hurt them as well as the enemy, so I tend to avoid this. Trying to get the timing right for lasers (to reduce shields) + beams (to do damage) is also kind of tricky if you're also monitoring a boarding party.

Also, it helps if you've got a couple of teams of boarders - while you're healing the first squad, send the second squad in to keep the pressure up.

Some ships are particularly bad for trying to board, however. If they've got significant crew, especially a lot of mantises, I may give up trying to win by wiping them out. On the other hand, getting into an engagement and realizing they have no medbay pretty much means I've got a win locked down.

As you've noticed, Rock and Mantis crewmembers are the best to send as boarders. Don't forget that you can pause the game with the space bar, so if you think getting your crew back on board is going to be a close call, pausing and unpausing to slow the game down may help you slice the timing razor-thin.

Possible spoilers if you've never been to sector 8:

It's also been noted in a couple of strategies for fighting the final boss that the weapon systems are in isolated rooms with just a single human guarding each. If you send 2 of almost any race of boarders into these rooms, you can kill the guard and disable the system, thereby making this fight easier. The best weapon to start with is (in my opinion) the missile launcher, which is third from the left.