Firstly, sodium can not form a compound with aluminium. They are both metals, and except for rare cases, metals do not form compounds with each other.
Secondly, the percipitate is probably Al(OH)3 like you suspected. Generally, Al(OH)3 is a fine whitish percipitate. However, if your electrodes were aluminium foil, then the carbon coating could make the percipitate look grey,
Next, depending on how long the electrolysis ran, your crystals could be NaCl or NaClO, assuming you used an undivided cell. I highly doubt you ran it long enough for there to be any appreciable amount of NaClO in the solution, and you probably would have said that it smells like bleach. So, I am assuming the crystals are NaCl.
Generally, using NaCl as an electrolyte is not a good idea due to the evolution of chlorine gas, which is VERY dangerous. If you let it go long enough in an undivided cell with NaCl electrolyte, formation of NaClO (and Cl2 gas dissolving in the solution) will turn it yellow.
So, if your solution did not turn yellowish, your crystals are probably NaCl.