What does the metaphor 'olive branch' mean in Pride and Prejudice in chapter 13?
Actually, the olive branch referred to in Chapter thirteen is more of a symbol than a metaphor. This is mentioned in the letter that Mr. Bennet receives from Mr. Collins, who is his nephew and who he has never met thanks to a disagreement that existed between them. However, Mr. Collins, especially as he is due to inherit Longbourne after the death of Mr. Bennet, now wishes to "heal the breach" between the two families, and the letter strongly suggests that he is thinking of marrying one of the Bennet girls to aid this process:
As a clergyman, moreover, I feel it my duty to promote and establish the blessing of peace in all families within the reach of my influence; and on these grounds I flatter myself that my present overtures of goodwill are highly commendable, and that the circumstance of my being next in the entail of Longbourn estate will be kindly overlooked on your side, and not lead you to reject the offered olive branch.
The olive branch is a well-known symbol of peace and thus is used by Mr. Collins here to express his intentions of making up with this side of the family that he has never met.
In addition to the answer above,
The Olive Branch metaphor would have been convective for the Bennets, because the situation in which the women would find themselves after their father dies will be quite precarious. For a peek on what could have happened to the Bennet sisters upon the death of their father, see what happens to the Dashwood sisters on Sense and Sensibility, where a similar situation (an entailment) occurs, leaving the the sisters destitute.
Hence, Mr. Collin's appeal for the Olive Branch is foreshadowing that his visit would be an unpleasant one, as it is a sign of what is to happen if the Bennet sisters do not find good husbands, and hence puts more pressure on the family as a whole.