What hope does Banquo maintain in Act 3, scene 1?
Here's the passage you ask about from Act 3.1 in Shakespeare's Macbeth:
Thou hast it now--King, Cawdor, Glamis, all
As the Weird Women promised, and I fear
Thou played'st most foully for't. Yet it was said
It should not stand in thy posterity,
But that myself should be the root and father
Of many kings. If there come truth from them--
As upon thee, Macbeth, their speeches shine--
Why, by the verities on thee made good,
May they not be my oracles as well,
And set me up in hope? But hush, no more. (Act 3.1.1-10)
Banquo's hope is, unfortunately, tied up with his suspicions. He knows about the witches' predictions, which makes him suspicious of Macbeth. This makes him a danger to the new king.
But the same witches that predict Macbeth will be king, also predict Banquo's heirs will be king. This is Banquo's hope.
Banquo is a bit ambitious, as Macbeth is. He would like to keep his name at the forefront of Scotland's leadership, and he would like to have his sons be kings. This is important in the context of the play because Banquo serves as a foil to Macbeth. Faced with a similar situation, though Banquo would like to see the predictions fulfilled, he is not willing to go to the lengths that Macbeth is to make them come true.
I think that the answer that you are looking for is that his descendants will be able to gain the throne of Scotland. He is hoping that they will be the kings after Macbeth. The reason that he has this hope is that the witches made their prophecy back at the start of the play. They said that he would not be king himself, but his descendants would be. Their other prophecies have come true (like when they said Macbeth would be the Thane of Cawdor and the king), so he hopes this one will too.