I don't know a lot about ESD, manufacturing or chemistry but I want to have a go at the following project:
I have a working laptop with a broken case. I would like to rehome the electronics in a case that I will make myself.
This will be a hobby and learning project. Obviously, I can buy a better and cheaper case than what I can make myself.
However, I am truly interested in doing this!
My first problem is ESD - What techniques are used in laptops to protect from ESD?
Similarly, what techniques do hobby electronics use?
Maybe it's all down to using special materials or I should have an ESD relief circuit covering my electronics?
I am looking at the existing case but I can see anything that might serve this purpose. I do notice the plastic doesn't make the fibers stand when I rub a sweater on it...
Natural wood will be the best and easiest to work with. Plywood, or boards of your choice. No pressed wood with plastic holding the particles together.
Paul
The biggest danger is if you develop a charge on your body and then you touch the circuitry and get a spark. Although, it doesn't take a lot of energy to fry an IC so you can get damage without a spark and without feeling anything.
Ideally, a [u]grounded conductive mat/surface and a ground strap for your body[/u].
The mat has high resistance so it doesn't short-out your electronics and the wrist strap has a high-value resistor so you don't get current through your body to-ground if you accidently touch a lethal voltage source.
The ground strap is most important. Often you can get-away with just touching ground (such as the case of a desktop/tower computer) before touching the electronic circuitry. But if you don't want to take any chances, use a ground strap!
You can also generally get away with a wood surface. Wood surfaces are slightly conductive and they usually won't hold a charge. But, wood usually won't "pass" an ESD test. Here at work we have some burn-in racks with wood shelves... Someday, QA will probably realize what we're using and then we'll get something else. 
...Our workbenches & test stations do have anti-static mats, and we have wrist straps & foot straps (and a specially-coated floor) and we have to test our straps every day.
But, I've done a LOT of electronics work without proper ESD protection and maybe I've only had an "unexplained failure" a couple of times. I'm NOT recommending that but I think the ESD thing is mostly overblown. It depends on where you live. I'm in California. We don't have extreme humidity, but I guess it's high enough that I rarely get "zapped" when I touch something. Also, some chips/components are more vulnerable than others and board assemblies tend to be more robust than single components.
What techniques are used in laptops to protect from ESD?
It's inside an insulated case. 
You could potentially damage it if your body has a charge and you touch an exposed connector.
ESD damage is cumulative. You may get by a time or two for each new device, but eventually the damage will be complete. I remember seeing a notice about the electronics on some hard disk drives being sensitive to ESD at 100 volts.
We do all you wrote about. Foot strap tested every morning and recorded. Our floor is bare concrete, cleaned each week with oil/sand. Concrete is also slightly conductive. I made wooden arm rests for the assemblers when they hand build SMT boards. And they are on ESD mats on the benches.
Paul
Plastic is the biggest culprit in ESD. So how do the plastic cases of our laptops and notebooks work? The plastic they use is infused with something conductive. Usually powdered aluminum because it won't corrode. They may also use carbon powder. If you use an ohmmeter to check if your old case is conductive, you will likely find it in the megohms range. (Static is kilovolts range so a few megohms is nothing to static charges). You can buy conductive paint to do the same thing to the inside of your new case, but it can be expensive. If you do decide to go with the conductive paint, be sure to run a wire or other electrical connection to the ground plane on the PCB.
ESD protection is on the circuit boards of the laptop already. It is applied to all connectors that reach the outside world. It is tested with a static discharge gun applied to everything metal poking out of the case. So it will not be necessary to add anything to your laptop.
The conductive coating applied to plastic cases is for EMI suppression in order to meet regulations. ( Electro Magnetic Interference ).
I am getting some really interesting info here - thanks all!
So, SteveMann vs. Grumpy_Mike - what's the verdict?
I mean, I am on the "do nothing" camp by default
PS: If I do go for doing something, how about this kind of thing?:
https://www.reichelt.com/gb/en/graphite-33-200-ml-graphite-based-conductive-paint-kontakt-218-p9480.html?PROVID=2788&gclid=CjwKCAjw4ZTqBRBZEiwAHHxpfq4t02ECe1usTTGhiVy-Vd0xFrlXrnHW_dP8c7qn-DTeUmuE5hD8ghoC0p4QAvD_BwE&&r=1
So, SteveMann vs. Grumpy_Mike
Well we don’t really do that sort of thing here. Only if you spot something you consider wrong do you attempt to correct another help giver.
Nothing wrong with what Steve said and nothing wrong with what I said. With these things it is a matter of degree.
I have been responsible for regulatory compliance at various companies both big and small so I have been there professionally. To cover all regulations both must be considered but he specific action taken depends a lot on what you find in an accredited test house and while testing with test equipment most amateurs will not be able to afford.
I have used plastic boxes without conductive coatings and they pass where as other things need them. Also often putting things in metal boxes makes emissions worse rather than better. It is somewhat of a black art.
Giving a simple answer is not possible in a forum. There is so much more you need to know and test.
oh, of course I was just kidding.
I've bought some conductive paints, carbon and copper. I will definitely have a play with ESD and EM pollution.