What custom does Hamlet mean when he says "But to my mind, though I am native here and to the manner born, it is a custom more honour'd in the...

Hamlet is referring to the custom of heavy drinking. King Claudius is not the only heavy drinker, but he sets an example which others are expected to follow. Hamlet strongly disapproves of it, mainly because it gives the Danes a bad reputation in other nations. When Horatio asks, "Is it a custom?", this suggests that Horatio is something of a stranger to the Danish court. He must live in some other part of Denmark. When Hamlet first meets him in Act 1, Scene 2, he says:

We'll teach you to drink deep ere you depart.

Horatio has been a fellow student at Wittenberg. Hamlet is wittily implying that Elsinore is an intellectual and cultural wasteland which has nothing to teach but heavy drinking. Hamlet is alluding to King Claudius, who is the heaviest drinker of them all. Claudius is seen drinking throughout the play, and he is drinking at the end when he accidentally causes Gertrude to kill herself with a poisoned drink intended for Hamlet and is then killed with the same goblet after Hamlet has stabbed him with the poisoned foil. The king's heavy drinking is probably meant to be a symptom of his guilty conscience. He is eaten up with guilt and fear of eternal damnation for having murdered his own brother. In Act 3, Scene 3, when Hamlet overhears him trying to pray, Claudius reveals that he is not the happy, carefree man he pretends to be.

O, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't,
A brother's murder! Pray can I not,
Though inclination be as sharp as will;
My stronger guilt defeats my strong intent....

Hamlet hates and despises everything about his uncle. He is especially annoyed by his uncle's drinking because he considers it disgraceful for a monarch to behave in public the way Claudius behaves. Hamlet tells Horatio that heavy drinking is indeed a custom,

But to my mind, though I am native here
And to the manner born, it is a custom
More honour'd in the breach than the observance.
This heavy-headed revel, east and west,
Makes us traduced and tax'd of other nations;
They clepe us drunkards and with swinish phrase
Soil our addition; and indeed it takes
From our achievements, though perform'd at height,
The pith and marrow of our attribute.

By "more honour'd in the breach than the observance," Hamlet obviously means that the custom of heavy drinking would be more honored by giving it up altogether. This is one way in which Hamlet shows a character superior to that of the King. Hamlet is the legitimate heir to the throne. Claudius is an unworthy usurper. Hamlet states his true opinion of Claudius to his mother in privacy in Act 3, Scene 4:

A murderer and a villain!
A slave that is not twentieth part the tithe
Of your precedent lord; a Vice of kings;
A cutpurse of the empire and the rule,
That from a shelf the precious diadem stole
And put it in his pocket!