The phrase "hold out olive branch" refers to making a peace offering or extending an invitation to reconcile a conflict. There are several synonyms for this phrase, including "extend an olive branch," "make amends," "offer an apology," and "make peace." Other similar expressions include "bury the hatchet," "mend fences," and "turn over a new leaf." These phrases all convey the idea of attempting to resolve differences and work towards a positive outcome. Using these expressions can help to diffuse tense situations and promote understanding and harmony between individuals or groups.