A benison is a blessing or benediction. In Shakespeare's play, King Lear announces that he has divided his kingdom into three shares to be given to his daughters as determined by their declarations of love for him.
In Act I, Scene 1, of King Lear, what does benison mean?
Goneril tells her father that her love for him is boundless. Regan says her love is even greater than Goneril's. Cordelia, however, replies that she loves him as a daughter should love a father, no more and no less.
Lear sees Cordelia's reply as rejection; in turn, he disowns Cordelia, saying that she will now be "a stranger to my heart and me."
In the meantime, the King of France and the Duke of Burgundy are suitors for Cordelia's hand in marriage. They are told that Cordelia will not receive a dowry or inheritance from her father. The Duke withdraws his suit, because a wife without a dowry is of no use to him. In contrast, the King of France claims that Cordelia is a prize, even without her share of Lear's kingdom, and announces his intent to marry Cordelia.
Here's how Lear replies to France's declaration for Cordelia:
Thou hast her, France: let her be thine; for we
Have no such daughter, nor shall ever see
That face of hers again. — Therefore be gone
Without our grace, our love, our benison. —
Come, noble Burgundy.