At an office Christmas party 21 years ago, when Elmer B. Staats was deputy director of outgoing President Eisenhower's budget office, he was given a seat cushion with a Republican elephant on one side and a Democratic donkey on the other.

"Elmer the Survivor" was the idea -- and the description fits Staats, the retiring head of the General Accounting Office. He served Truman, Eisenhower, Kennedy and Johnson in the budget office post, before becoming GAO head under Johnson, Nixon, Ford, Carter and briefly, Reagan.

Another longtime acquaintance remembered that an budget director never scheduled an early-morning meeting with Staats because Staats' attention to detail would surely push the meeting past the lunch hour. That's another description that fits Staats: Elmer the Hardworking Professional, the Consummate Bureaucrat.

For the last 15 years Staats has practiced his hardworking, low-key, professional style of management as comptroller general of the United States, head of the auditing and investigative arm of Congress. He had kept members of both parties happy, a sign of evenhandedness, according to admirers, a sign of timidity, according to critics.

He has taken the GAO from the world of the straight financial audit to the more complex role as evaluator of government programs. A few years ago he even set up a fraud "hot line" which attracted thousands of calls from citizens reporting potential cases of abuse of federal programs.

On Friday, after a full term at GAO and 42 years in federal service, the 66-year-old Staats officially retired. Aides said he was having trouble letting go. After all, he had helped set up the executive office of the president in 1939, and has had a unique vantage point for watching the growth of the federal government ever since. "I'll probably be in some next week," he said.

The general respect with which Staats is viewed was evident before the House Budget Committee last Tuedsay, his last appearance before Congress. His testimony was interrupted several times by laudatory comments from members of both parties. An example from Rep. Paul Simon (D-III): "You represent what the American people yearn for in a public servant."

The GAO, which now has 5,000 employees and is asking for a $244 million budget in fiscal 1982, has a glittering reputation in most circles. It's blue-covered reports often are headline-making tools for an eager press and members of Congress, whose requests make up nearly 40 percent of GAO's workload.

But the agency is not free of critics. In recent interviews, executive branch and congressional officials had these complaints:

Reports at times take so long to complete and are so watered down in GAO's extensive review process they are of little value. Occasionally reports have even been killed, apparently in the face of congressional opposition.

For example, a few years back, a report critical of the Army Corps of Engineers' cost-benefit ration on the Tennessee-Tombigdee Waterway was stopped after Sen. John Stennis (D-Miss.), a backer of the project, complained. Staats said a court suit stopped the report, but the order didn't come until after Stennis complained.

In churning out nearly 1,000 reports a year, the GAO is stread too thin and its evaluators don't have the program or sampling expertise to make valid analysis in some cases. At the same time, a few areas like the intelligence community are immune from GAO monitoring.

The agency is not aggressive enough in following up its recommendations to see if executive departments are correcting problems. Staats admits agencies often "hope we'll forget them."

The GAO has shield away from monitoring the work of Defense Department contractors ever since Staats' predecessor was savaged in House hearings in 1965 for being too ambitious in tracking cost overruns. For instance, it was a whistleblower, not the GAO, who disclosed the cost overruns on Lockheed's C5A airplane in the late 1960s.

But as one veteran Capitol Hill aide put it, "You have to grade them on the curve. They're the only game in town."

Staats acknowledges that there is room for improvement at the GAO -- and he has taken the initiative to respond to criticism in several areas.

Two years ago, he began to hear complaints about the quality of the analysis in some of his reports. "We realized we needed to do more to strengthen the methodology that goes into these reports," he said during an interview in his last week in office.

So he set up a training course and an in-house institute of experts on statistical sampling and systems analysis. Richard Fogel, head of GAO's office of program planning, said, "You've got to give the guy credit. He surfaced that issue and we're trying to deal with it."

The agency also is trying to improve the timeliness of its reports, which often are more than a year in the making. As for complaints about lack ofr aggressive follow-up, Staats noted that follow-up on the recommendations in GAO reports ultimately falls back on the leadership of the executive branch and Congress.

As he looked back at the changes he has seen in the federal establishment. Staats emphasized the need for a stronger commitment to the management of federal programs. "We have to put the M back in OMB," he said, refferring to the Office of Management and Budget.

For too long, he said, the emphasis in government has been lopsided toward delivery of service and money, with too little attention paid to the business side, Thus, the Treasury is losing billions of dollars a year because delinquent debts owed the federal government go unpaid, and challenged disbursements remain unresolved.

"I thingk the leadership in this are is going to have to come from the very top in OMB and the White House," he said. When David A. Stockman was appointed head of OMB in December, Staats fired off a letter of congratulation that included a suggestion that he establish a deputy for organization and management "to give this side of the Omb sufficient stature."

He said the management role of OMB had been neglected inrecent years. "Indeed, I would go so far as to say that without a stronger management leadership role OMB will not be able to achieve all of the financial savings which it has an opportunity to save," he wrote.

In an interview and later in testimony before the Budget Committee, Staats also took issue with the Reagan administration's decison to reject changes in the indexing of major entitlement programs and to impose a new

Noting that indexed programs, such as Social Security payments and civil service and military pensions, now account for nearly one-third of the budget. Staats said: "This is a fairly fundamental point. We don't understand why the new administration has backed off from any effort to put some restrictions on it [indexing]."

He said he thinks the hiring freeze is "the wrong way to go about making savings in personnel costs." He suggested putting a limit on the dollars spent for government jobs, not limits on the number of people. "That would give the manager some flexibility," he said.

Despite some criticism that the GAO has been timid in certain areas, such as monitoring the work of defense contractors, other say Staats has not backed down from issuring reports with strong recommedations. For instance, a GAO report suggesting repeal of the Davis-Bacon Act, requiring union-scale wages on federal construction jobs, upset several pro-labor members of Congress.

"It's a fine line we have to walk," the GAO's Fogel said. "But most people in Congress respected our independence. They leave us alone."

Rep. Jack Brooks (D-Tex.), chairman of the House Government Operations Committee and a longtime observer of the GAO, said Staats has done an outstanding job in heading the key in Congress.

Now Brooks and a special bipartisan commission of congressional leaders are sifting through names of suggested successors to Staats. Under a new law, they will submit three names to President Reagan, though he is not bound to limit his choice to the list.