Th political star of Israel's Defense Minister Ezer Weizman is on the rise. Once thought of as a bit of a playboy and a political lightweight. Weizman has been quietly and systemtically picking up political support and a new national reputation as a serious leader.

A recent public opinion poll showed that 71.6 percent of those queried expressed satisfaction with Weizman as defense minister while only 68.4 percent expressed the same confidence in Menachem Begin's performance as prime minister.

The same poll registered public approval of Begin's performance at 78.3 percent in December.

The polls were not comparative and should not be taken to indicate that Weizman is more popular than Begin. If Israelis were asked whom they would rather see as prime ministe, Weizman would not come within hailing distance of Begin on anybody's survey. But there is evidence that Weizman is gradually establishing himself as a logical choice as the number two man in the Israeli leadership. The thought that Weizman might take over should Begin have to step down for health reasons no longer sends as many Israeli eyes rolling heavenward as it did half a year ago.

The two mainstays of Begin's cabinet are Weizman and Zalman Shoval, a member of the executive of the Likud Party, to which both Begin and Weizman belong, expressed the thoughts of many when he said recently: "Ezer is certainly creating a good image as minister of defense although he still may have to prove his statesmanship."

Although Weizman and Begin are both running into flak from the right wing of the Likud Party because of Begin's peace plan, the fact that Weizman is the number two man after Begin in the Likud's Herut faction, the largest faction within the ruling party, may give him an edge over Dayan if it ever comes down to a question of succession. Dayan is not even a member of the Likud and having crossed from the opposition Labor Party to join the government, his political base is splintered and comparatively unorganized.

Weizman, a nephew of Israel's first president, Chaim Weizman, was a former Royal Air Force pilot during World War II who rose to become head of the Israeli air force and the architect of the air force's stunning performance in the 1967 war. For all his tactical brilliance, Weizman acquired the reputation of a hard drinking, swaggering, Errol Flynn type who flew a black-painted Spitfire in his spare time and shot his mouth off in public all too often.

"He had the reputation of being a sharp-tongued, super hawk wise cracker who was too blunt for his own good," said one government official.

All this has begun to change. Almost as soon as he took over the Defense Ministry in June, Weizman adopted a policy of keeping quiet and concentrating on his new job. The old bombastic, simplistic rhetorical style began to fade and in its place emerged a thoughtful moderate, responsible leader with a good grasp of the problems facing Israel.

It is not unusual for a man to change when given the responsibility of high office but personal friends say he became far more serious-minded and introspective when his son was badly wonded during the war of attrition in the early 1970s.

Weizman was the architect of the Likud's upset victory in the May elections and virtually ran the show after Begin suffered a heart attack just before the election. He began to gain international stature during the Lebanese crisis of last autumn when he was dealing with the Lebanese and the Americans to bring about some sort of cease-fire between the Israeli-backed Christian forces in Southern Lebanon and the Palestinians.

Weizman's new image improved when he and Egypt's President Anwar Sadat established an immediate raport during Sadat's visit to Jerusalem. Wa it spontaneous or did Sadat plan it? It is known that Sadat made a careful study of Israel's leaders before he made his dramatic move in November. Perhaps he sensed in Weizman a man with whom he could deal.

Certainly Weizman carries less emotional and political baggage than Begin when it comes to Israeli settlements in the occupied territories and Israeli's biblical claims.

On the other hand, even his critics concede that Weizman has a great deal of personal charm and wit. Whatever the case. Weizman was the first Israeli leader invited to meet with Sadat on Egyptian soil, and has had more direct dealing with the Egyptians than Dayan or Begin.

Weizman has his detractors. Haqqai Eashed, political analyst for the newspaper Davar, says that Begin ha s a problem with Weizman who "likes to present himself not only as an heir but also as prince consort." Eashed also criticizes Weizman for trying to be too much Sadat's "fair haired boy."

Weizman tried to challenge Begin's leadership of the Herut faction in the early 1970s and got his wings severely clipped. There is little evidence that he would risk this again and members of Begin's staff say his relations with the prime minister are excellent.

There is little doubt that present foreign policy is run by Begin and Dayan. But Weizman is gaining new respect both within his party and the country at large and may yet emerge as Begin's heir.

Ironically, Weizman may have to fall back on former Labor Party and liberal elements within the Likud for support as the Herut faction continues to drag its feet in opposition to Begin's peace plan. there are doves and moderates within the government coalition who are already saying that Ezer Weizman as prime minister may be the best chance of making peace with the Arabs. No one would have said that a year ago.