The U.N. Security Council takes decisions on several issues that affect Saudi foreign policy interests, most notably Syria and other issues on the Middle East and Iran. This should give Saudi Arabia an interest in influencing council decisions. Yet as a non-permanent member, Saudi Arabia would have little power to affect votes given that five states (the U.S., China, France, the U.K. and Russia) have veto power (the ability to block any resolution). Non-permanent membership confers perhaps some greater ability to shape the agenda, but important states in an affected region are generally consulted anyway when major decisions are taken.
Some states trade their votes in the U.N. Security Council for U.S. aid or even favorable World Bank decisions. Saudi Arabia is not in the market for either. Moreover, as Michael Ross and I have shown, states rich in oil are less eager to bind themselves to and participate in international organizations.
All of this would matter little if there weren’t an important downside to U.N. Security Council membership. Any time the council deals with a major crisis, any non-permanent member is forced to publicly take a position. This often presents a problem, especially for states who depend on the U.S. even though the U.S. is unpopular domestically or regionally. Or, simply because the U.S. has different foreign policy interests. As the map below shows (see explanation here), Saudi Arabia often votes against the U.S. on U.N. General Assembly resolutions. But those are symbolic resolutions. It can sometimes be very convenient to remain ambiguous when things really matter. Saudi Arabia depends heavily on the U.S. for military equipment. Why upset the U.S. if there is little to gain from a seat on the Security Council? Indeed, Saudi Arabia has never before been a member of the UN Security Council!
Perhaps they initially sought a seat this time because there are so many issues at stake that matter to them. Why then did they change their minds at the last moment? One speculative hypothesis is that they were impressed by the U.S. cutting off the delivery of large-scale military systems to Egypt, suggesting that the threat of aid removal has become more credible. The increased credibility of that threat may be enhanced by increasing energy production in the U.S. The expected cost of getting into serious trouble with the U.S. may just have gone up enough to make participation in the Security Council simply no longer worth it.