The surname "ALKOK" seems to have it's roots in Saaremaa, Estonia sometime around 1732. The meaning is unknown and therefore one theory has been that it originated from a British sailor that had been shipwrecked off the coast of Saaremaa. Thus the name "Alcock" was changed to "Alkok". Other than a relative's belief in this, there is no proof.
More likely, the name came about the same way most Estonian surnames were created: referring to something in nature or a particular profession of the individual. Even then, the name "Alkok" is unique in that there is no clear meaning. It could be a chef as in "kokk" or if the name was originally "Alkukk" as in some early documentation, it could be referring to a rooster or "kukk". The "Al" portion of the name would be a mystery.
The Alkok genealogy appears to go back to the early 1700's where a person lived and worked on a farm. The idea of either a chef or the rooster would therefore not be far-fetched.