Firstly, the mediator should be a third party everyone can trust, such as United Nations. I hope that the UN presence in Lebanon will give it more influence in Middle East and in general. In addition, United States can help by putting some pressure on Israel about working to get peaceful solution to the conflict. It is needed to get Hamas and Fatah to the negotiation table as part of a negotiation body the other side cannot ignore. This way, the two organizations will be held responsible for the palestinian acts in the peace process and thus cannot act in irresponsible fashion. Following will be needed for peace:

-Israel to stop oppressing the Palestinian population.
-Israel to withdraw its presence in Gaza and West Bank.
-Israel to withdraw from Lebanon(in order to have peace with Hezbollah, which is a separate faction in respect to the Palestinian factions)
-Suicide bombings and other Palestinian terrorist attacks to stop.
-Hezbollah to stop attacking Israel(Which it has agreed to do if Israel withdraws).