What function does the friendship of Hamlet and Horatio serve in the play?

1) The guards' knowledge of the friendship prompts the decision that gives the ghost legitimacy. The guards have Horatio clarify the ghost as a real essence--not just a figment of Hamlet's imagination.

2) Drives Horatio's desire to redeem himself as a worthy friend to Hamlet.  On the roof, Horatio so scares himself with his story of omens at caesar's death that he acts irrationally to the ghost, losing the apparition and his chance to clarify the ghost's appearance ahead of time for Hamlet.  Horatio's guilt for abandoning this duty to Hamlet moves him to stay by Hamlet for the rest of the play. 

2) The friendship serves to validate Claudius' guilt--Hamlet turns to Horatio to second Hamlet's interpretation of the King's response to the play.

3) The friendship provides for Gertrude's sole account of Ophelia's death--again, in Horatio's desire not to abandon Hamlet, Horatio chooses to meet with Hamlet after reading Hamlet's letter.  In doing so, he leaves his watch over Ophelia, so the only evidence of Ophelia's death resides in Gertrude's story.

4)  The friendship provides for evaluations of Hamlet's moral character--the extent of Hamlet's moral degeneration emerges in Horatio's shock over Hamlet's cold-blooded response to Ros. and Guild,'s likely inability to repent before they die and also in Horatio's lamentation of the extent of Hamlet's moral fall after Hamlet dies: "Now cracks a noble heart."