President Reagan last night said he will nominate Elizabeth Hanford Dole, a member of his White House staff, to head the Department of Transportation.

She would succeed Drew Lewis, who resigned last week to take a job in private industry.

The president praised Dole, who stood at his side at the beginning of last night's news conference, saying she "has been performing magnificently as my assistant for public liaison at the White House."

Faith Ryan Whittlesey, a longtime Reagan political activist and now the ambassador to Switzerland, will replace Dole at the White House, Reagan said.

Dole, 46, would be the second woman in Reagan's Cabinet, joining U.N. Ambassador Jeane J. Kirkpatrick, and the first to head a department.

Reagan has long been urged to name more women to his Cabinet. The demand came originally from women's organizations, and has been supported by White House officials, including chief of staff James A. Baker III, and by Republican officials eager to improve the president's standing with women voters.

Asked by a reporter as he was leaving the East Room whether he had become "pragmatic" and would now support the Equal Rights Amendment, Reagan replied, "What we have been doing has far more meat to it than the ERA."

Dole, a former member of the Federal Trade Commission and the wife of Sen. Robert J. Dole (R-Kan.), brings the assets of Washington experience and virtual certainty of Senate confirmation to the job. Although considered effective at the FTC, Dole has not been notably influential within the White House staff. Her supporters within the administration say this is because she has not been given the opportunity to show what she can do in the largely male environment of the White House.

She will have this chance at Transportation, where complex pieces of new legislation need a skilled administrative hand and where many of the department's ranking subordinates are regarded as less than first-rate by the transportation community.

Lewis left a legacy of adequate funding for such major programs as highways, airports and development of the air traffic control system. But his successor faces thorny issues of deregulation and the problem of rebuilding an air traffic control system damaged in the controllers' strike.

Dole's nomination may be a clue that the president is listening to his advisers more than he has indicated in his public statements. Senior White House advisers have generally favored broadening the outlook of the Cabinet by bringing in women and minorities.

But Reagan, never quick to make changes among his top-level appointments, has been reluctant to shuffle the Cabinet.

In the case of Transportation, his hand was forced by Lewis, who left to become chief executive officer of Warner Amex Cable Communications Inc. after it became clear that no other position in the Cabinet or White House staff was likely to become available.

Since Lewis' departure was well-advertised in advance, several prospective successors made their desires known to high-ranking members of the administration. These included Deputy Transportation Secretary Darrell M. Trent, Federal Highway Administrator Raymond A. Barnhart, former representative Thomas B. Evans Jr. (R-Del.) and Los Angeles County Supervisor Pete Schabarum.

One White House official suggested last week that the president go outside the administration and name a "star" to fill the Transportation post that becomes vacant Feb. 1. He mentioned Chrysler Corp. President Lee A. Iacocca, who said later that he intends to remain with Chrysler.

Among White House insiders and others close to the administration who would like to see more women in the Cabinet, Dole was the clear favorite.

Dole grew up in North Carolina and graduated from Duke University with a major in political science. She graduated from Harvard Law School as a Phi Beta Kappa.

In the Nixon administration, she served as aide to Virginia H. Knauer, the president's assistant for consumer affairs. In 1973 she was named to a seven-year FTC term.

In December, 1975, she married Sen. Dole, who had been divorced. When he became Republican vice presidential candidate less than a year later, she resigned from the FTC to avoid any appearance of conflict of interest.

She earned praise from Republican officials for public appearances in that campaign, and appeared on occasion as a stand-in for her husband when he was a presidential candidate in 1980.