SEIZE THE MOMENT of broad and fresh interest in the Palestinian issue and convert it into progress toward an Israeli-Palestinian settlement, says Egyptian president Hosni Mubarak, visiting Washington. But how? Can something useful be done?

The Mubarak suggestion in which his American hosts apparently showed the most interest was a call for a six-month cooling-off period by Israelis and Palestinians. The proposal is not as innocent and uncontroversial as it may seem. Assume that responsible leaders on both sides could make it happen. Why would Palestinians wish to abandon the tactic that has brought them their greatest gains in public opinion in 20 years, without being assured access to a political forum to press their claims for self-determination and statehood?

In fact, Mr. Mubarak has an answer to that. He thinks that the PLO can be brought to a safe two-state peace by such Arabs as himself, and he thinks that half the Israelis are ready for peace and the other half can be persuaded by the United States if it is so minded. Rather than wait for all ducks to be put in a row -- a project that risks becoming an alibi for dodging the tough questions -- the Egyptian leader wants to call the long-pending international peace conference, charge the Arabs with finding an acceptable way to tuck in the Palestinians, set up a working group on the Palestinian-Israeli issue and let it proceed. Why wait for the Israeli or American elections or anything else? he asks -- just do it.

Any Middle East adviser worth his salt can give his chief a dozen respectable-sounding reasons to go slow or say no. President Reagan, for one, apparently hesitates to take up a major, difficult and risky regional initiative at this late and otherwise full moment in his presidency.

But powerful new considerations, in addition to the powerful old ones, argue for a decision the other way. There is a growing awareness of the political aspect of the Palestinian dilemma. There is a growing clarity to the American view of why Israel needs a settlement and why it needs to be assisted to a settlement. To the deep and continuing American commitment to Israel's security is being joined a new questioning of the United States' inadvertent role as cosponsor of Israel's occupation. A president prepared to take a responsible lead could expect not only substantial international cooperation but also broad support across the American political spectrum.