Polycarb is a definite improvement over nylon - in terms of impact strength. Polycarb, referred to most often by its trademark in the US, 'Lexcen', is the corner stone of 'bullet resistant glass' among other things. Nylon on the other hand has superior mechanical properties - i.e, it can flex without failure...
Ultimately - there is no 'answer' to your question .. You choose what is either A: On hand (most common) or B: What ends up with the preferred presentation of the part and finally C: what meets the mechanical strength requirements of your specific part. 90% of people will go with option A with consideration of B... if you're doing something specific then you will look at C.
A is easy - what have you got.
B is personal choice combined with the engineering requirements - If you want translucent glass-like look, you might use T-Glasse, if you want glow-in-the-dark you can choose PLA or ABS... if it has high surface detail then PLA wont be a good choice.. etc...
C again is something only you can answer for the part you want to make. As already alluded to:
PLA is 'organic' (not to be confused with 'eco friendly'; as it contains secret sauces) and is Brittle - it will exhibit a hard 'shell' compared to ABS, but will shatter and fail very quickly with any tensile loading.
ABS is flexible to an extent, so has superior mechanical properties over PLA - but it also has much more stringent tolerances for good layer adhesion and final bond strengths and its a trade off - High temps bonds layers well, but reduces the impact resistance...
HIPS is used mainly for Dissolveable Support Material - however - it can be argued HIPS exhibits high impact strength than ABS - If I was making a 'presentation' case I might use ABS and all its glorious colours - but If I was making a case to be thrown around in a tool bag.. I might just as likely use HIPS as its acetone and many otehr solvents resistant and is a lot more resistant to impact against the threads.
Nylon comes in a couple of flavours... 618 is good for quick turn prototypes or casting molds. 645 offers slightly better bonding qualities and much higher mechanical strenght, comes in clear, and is appropriate for making bones or other 'industrial' nylon application models.. And then there's T-Glasse; which ultimately is more about presentation than anything else. (618 with good finish). However - Nylon will not bond with anything else but Nylon - so parts need to be bonded with mechanical fixtures and can't be 'glued' or any such... But then nylon also display an extremely low friction co-efficent, meaning it makes great bushing/bearings/gears/pulleys etc...
Polycarbonate has the highest impact resistance; but much like carbon fiber, its dependent on where those forces come from... It is also arguably the most chemically resistant whilst still being bondable...
Laywood/Laybrick - well, they are options 
I'll leave those ones up to the reader...
Anyway - basically - there's no 'magic formula' to choosing the 'ideal' filament - its a question of evaluating what the requirements are, and then assessing what works best for those requirements. They can often end up being mutually exclusive - e.g, 'needs to be slippery' combined with 'needs to be glued together due to model complexity' means nylon is both suitable and unsuitable. Live without nylon, or redesign the part.. etc.. Is the part going to live outside or in a window where sunlight will hit it - Then PLA is out due to its lack of UV stability....
You need to read and understand the material data sheets for each filament type, and asses that against the requirements of the specific model and its specific applications... and that's something you (not you you, the royal you ) need to do yourself as only you know what is and isn't an acceptable trade-off.
