President Carter yesterday chose Robert S. Strauss, one of his most trusted all-around troubleshooters, to head the U.S. mediation effort in the next stage of Middle East peace talks between Egypt and Israel.

In that position, Strauss will be responsible for the administrationhs drive to move beyong the Egyptian-Israeli peace treaty and bring the two countries into agreement on a self-governing mechanism for the Palestinian inhabitants of the Isreali-occupied West Bank and Gaza Strip.

A solution to the Palestinian question is crucial to the administration's hopes of getting other Arab countries to support the Middle East peace effort. In naming Strauss as ambassador-at-large to handle the U.S.-mediated talks, the president undrescored anew the high priority that he has assigned the drive for a comprehensive peace in a region.

Strauss, who currently is serving as Carterhs international trade negotiator, has emerged as one of the small circle of White House insiders relied upon by the president for highly sensitive tasks. Previously, Strauss headed Carter's campaign to arrest inflation and, as a former chairman of the Democratic National Committee, he has been one of the president's most influential political advisers.

The wealthy Texas lawyer, who is noted for his skills as a colorful and persuasive conciliator, now faces a task that is expected to be even more difficult and time-consuming than the five months of cliff-hanging negotiations that were required to produce last month's signing of the accord between Israel and Egypt.

Israeli Prime Minister Menachem Begin faces heavy pressure within his country either to retain control over the West Bank and Gaza Strip or to keep Palestinian autonomy within tightly constricted bounds that will prevent the emergence of an independent Palestinian state.

On the other side, Egyptian President Anwar Sadat's separate peace with Israel has estranged him from his traditional allies in the Arab world. His ability to regain influence with such key Arab states as Saudi Arabia and Jordan will depend directly on his ability to get a settlement for the occupied territories that is perceived within the Arab world as satisfying Palestinian demands for a homeland.

These conflicting pressures were obviously very much on Carter's mind when he appeared before reporters with Strauss and Secretary of State Cyrus R. Vance yesterday to say:

"The treaty is more than a monument to past efforts. It's a promise of a comprehensive peace in the Middle East. That promise must be kept."

It was for that reason, Carter added, that he asked Strauss to take on the job of mediating the Middle East talks. The president added that he spoke by phone yesterday with Begin and Sadat and said that both leaders were "positive and enthusiastic" about Strauss's appointment.

Strauss, who normally employs a hearty and jocular manner in talking with the press, was uncharacteristically solemn in his comments. He said:

"I suspect that of all the tasks I have undertaken this is by far the most complex and difficult and the one that offers the most rewards."

He recalled that on Monday night he had been musing aloud with his wife about the many jobs he has held in public and private life, and added: "I am reasonably well convinced that these were just in training for this assignment."

In response to questions, Strauss noted that the West Bank-Gaza talks are expected to start at a relatively low technical level and that it probably will be the summer before be becomes involved in them full time. In the interim, he and Carter noted, Strauss will concentrate primarily on the effort to get the administration's foreign trade legislation through Congress - a task that the White House hopes to complete by the time Congress goes into summer recess.

Until now, Alfred L. Atherton Jr., a career Foreign Service officer, has been serving as special ambassador for the Middle East talks.Atherton is expected to become ambassador to Egypt, but reliable sources said yesterday he will remain in Washington until Strauss is able to shift full-time into the mediator's post.

In an echo of the problems Strauss will face, the State Department yesterday charaterized as "regrettable" Israel's recent decision to create two new settlements of the West Bank and said the action will make negotiation of the Palestian issue "that much more difficult."

The Isreali decision, announced Sunday, was made over the objections of the United States, which considers all settlements in the occupied territories as illegal. However, yesterday's State Department statement was much more low key and restrained than in the past when the administration criticized new settlements in very strong language.