Tough Little Ship is a Ship Achievement for the Kestrel Cruiser.
"As the Kestrel Cruiser, repair back to full health when it only has 1 HP remaining."
For the game to recognize this achievement as earned, the player must have the Kestrel Cruiser down to one hull point, and then the ship must be fully repaired all at once at a store beacon. If the ship is not repaired all at once or if the Hull Repair drone is used instead of a store, the achievement will not be earned. Having 1 HP remaining when meeting the Rock war vessel for the third time will earn the achievement as well.
To earn the achievement somewhat more honestly, the most reliable method may be to obtain enough scrap to heal the Kestrel to full (usually 58 or 87 scrap), then find an asteroid field near a store. By going to the asteroid field, destroying any enemy that might be there, then powering down the shields and letting the asteroids hit the ship will take the ship down to one hull point fairly safely. Then just a quick warp out to the store should unlock the achievement.
A somewhat safer though slower way to do this revolves around the Fire Bomb. After obtaining the necessary scrap to fully repair the ship (and ensuring there are several missiles handy), having the ship damaged down to five or so HP. You can do this in a store beacon. Then the Fire Bomb may be used on the player's own systems and subsystems. Whenever a system is destroyed it will do one point of damage to the hull. Once the ship is at one HP, the fires should get extinguished and then repair at the store will earn the achievement.
What's probably the safest and slowest way is to use a fire to repeatedly destroy a single system, doing 1 point of hull damage each time, to very slowly whittle your ship down to one hull point. It's very simple to set up because all you need to do is keep a empty room on fire (this requires level two oxygen) while you go next to a store, using a store you already found should work. Then you can have the fire spread into a room with a system destroy it, put out the fire, repair one level of the system and repeat as needed.