At the beginning of Hamlet Marcellus, Bernardo, Fransico, and Horatio are standing guard. They have seen a ghost the night before and are hoping that he will appear again this night. Marcellus speaks when the apparition appears "Thou art a scholar; speak to it, Horatio." Marcellus knows that Horatio is Hamlet's most loyal and trusted friend, and so he implores Horatio to speak to the ghost. Based on the ghost’s clothing (his armor) the guards believe that the ghost is Hamlet’s dead father, but in order to be sure, someone needs to talk to him and make sure. The logical choice would be to have the best friend of the ghost’s son speak to him first.
However, when Horatio speaks the ghost goes away. We soon learn that it only his son he wishes to speak with.