What hope does Banquo maintain in Macbeth?
I believe that Banquo, in Shakespeare's tragic play Macbeth, possess many different hopes.
First, Banquo hopes that the prophecies are correct. The prophecies were correct when it came to Macbeth's future as the Thane of Cawdor and his gaining of the crown of Scotland. Therefore, Banquo surely hopes that the part of the prophecy about his own sons gaining the throne will come true as well.
Second, Banquo seems to hope that Macbeth held true to his statement about chance. Upon hearing the prophecies, Macbeth makes an aside (a quiet statement made aloud to himself).
If chance will have me king, why, chance may crown me,
Without my stir (I, iii, 143-144).
One could assume that Banquo heard the quiet comment based upon a statement later made by Banquo.
Thou hast it now: King, Cawdor, Glamis, all,
As the weird women promised, and I fear
Thou play'dst most foully for't (III, i, 1-3).
By stating that he (Banquo) fears that Macbeth "play'dst most foully for't," Banquo does hold some hope that Macbeth allowed chance to crown him (based upon the fact that he does not outright accuse Macbeth of the murder of Duncan).
Lastly, Banquo hopes that Fleance will survive Macbeth's murderous rampage and enact revenge upon Macbeth. This is seen in Act III.
O, treachery! Fly, good Fleance, fly, fly, fly!
Thou mayst revenge. O slave (III, iii, 17-18)!
This hope is defined in the fact that Banquo states that Fleance must become a slave to the revenge upon Macbeth. Therefore, Banquo hopes that Fleance will have revenge upon Macbeth.