Not really. However, this is not Mexico's fault. Mexico just happened to be elected into one of the most ineffective international institutions of the 20th Century: the UN Security Council. Many could argue that the Security Council is great at investigating, recommending, formulating, determining, calling, and exercising a number of functions. Unfortunately, the Security Council has clearly failed in maintaining peace (or avoiding war) and this is its most important defect. Mexico is now part (again) of that defective machine.
But suppose that the Security Council did not avoid conflict and it is now ready to issue a resolution (12 days later after the initial confrontation and with 770 Palestinians and 14 Israelis killed in action in the case of the conflict between Israel and Gaza). Is Mexico important in this case? It depends. Mexico's vote matters in irrelevant issues‚ like a resolution that demands parties to stop fighting. Mexico's vote, like Japan's or Burkina Faso's, does not matter in substantive matters. As indicated by the Security Council's Website: "Decisions on substantive matters require nine votes, including the concurring votes of all five permanent members." Thus, if the big five have not agreed on something, then the votes of the non-permanent members do not really matter.
But Mexico and the Mexican Ministry of Foreign Affairs should not be too concerned about their lack of influence. Since Mexico contributed to the peace agreements in Central America in the 1980s, it has not exercised any major role in the international system (drug trafficking may be an exception). By now the country and its representatives to the UN should be used to having small roles in substantive international politics. Sadly, things are not likely to change in the future.