Frank M. Johnson Jr., a federal Judge from Alabama who has a reputation of being tough with criminals as well as a strong supporter of civil rights, was nominated as expected yesterday to become the new director of the FBI.

Attorney General Griffin B. Bell made the announcment at the White House, saying he turned back to the administration's first choice for the job after he learned recently that Johnson thought he "made a mistake" in rejecting President Carter's offer in December.

Bell said he spent two hours discussing the FBI and its problems with Johnson at a Holiday Inn in Newnan, Ga., on Sunday and then passed on his recommendation for Carter's approval Monday.

The rapid turn of events ended a seven-month administration search for a person to lead an FBI that has been plagued in recent years by revelations that it spied on and harassed many American citizens, including some in civil rights groups and other domestic protest movements.

Johnson said at a press conference in Montgomery yesterday that he was honored by his selection and hoped "I can be as fine a leader as J. Edgar Hoover."

He said he had been unable to accept Carter's earlier offer because of the illness of his mother, whose health has now improved.

In picking Johnson, a life-long Republican who is chief judge of the U.S. District Court in Montgomery. Bell and Carter passed on a list of five candidates recommended by a blue-ribbon search committee.

Bell said, though, that Irving S. Shapiro, the Du Pont Co. board chairman who headed the panel, was satisfied with Johnson's selection. In fact, immediated reaction to the choice was almost uniformly favorable.

One official at FBI headquarters in Washington said he hadn't heard a single negative comment about Johnson after talking to more than a score of special agents and administrators. Another veteran FBI agent said he welcomed Johnson's appointment because "he seems to have the credentials and the clout to stand up to some of the people who have been walking all over us."

Key members of Congress also greeted the news effusively. Sen. Edward M. Kennedy (D-Mass.) a member of the Senate Judiciary Committee which must confirm the FBI director, said he welcomed the judge as "a man of utmost integrity, competence and courage."

Rep. Don Edwards (D-Cal.), a former FBI agent who is chairman of the House Judiciary subcommittee with FBI oversight, said Johnson was "a towering figure in the field of justice who definitely has the experinece and the judgement for this job."

Sen. James O. Eastland (D-Miss.), chairman of the Senate Judiciary Committee, declined comment on the selection.

Eastland's committee is expected to start confirmation hearings soon after Congress reconvenes in September. But Bell said Johnson would not officially take over his new post until the current FBI director, Clarence M. Kelley, retires Jan. 1.

Kelley issued a statement yesterday saying Johnson's service as a U.S. attorney and federal judge "prove his dedication to our criminal justice who definitely has the experience and the judgme known for his landmark rulings that desegregated Alabama schools and prisons and guaranteed mental patients' rights to treatment, he is also known as a judge who is tough on criminals.

"He's a law-and-order judge," said Ira DeMent, the U.S. attorney in Montgomery."If people need to go to the penitentiary, he's not afraid to send them. The sentences are certain."

Even those who opposed Johnson in court joined in his praise yesterday. Gov. George C. Wallace of Alabama, a law school classmate who later became a political enemy, said, "We have had our differences, but I never considered them personal. He is a hard worker and I wish him the greatest of success . . ."

Maury D. Smith, a lawyer who represented the state of Alabama in several of the key cases decided in Johnson's court, said yesterday. "He's one of the most thoroughly capable people I have known. There's no person I respect more than Judge Johnson."

Bell said at the press conference that Johnson had pledged to him that he would serve the full 10-year term as FBI director. He emphasized that there were "no side deals," such as the enticement of a future Supreme Court seat, to lure Johnson away from the Alabama judgeship he has held for 22 years.

Bell noted that Frances M. (Kelly) Green, a deputy associate attorney general and a former law clerk of Johnson's, told him a few weeks ago that the Alabama judge had changed his mind about being considered for the top FBI post.

Bell said he then arranged for the meeting in Georgia last Sunday, and that Johnson told him then that he would look forward to the new challenge of leading the FBI.

Bell said that he considered his recommendation of Johnson the most important he would have to make as Attorney General. "I wanted to find someone the bureau "particularly the younger people in the bureau, would think was a compliment to them, someone they would be honored to have as a leader. I think Judge Johnson is that sort of a person."

The sensitivity with which the Carter administration viewed the FBI nomination is apparent in a letter that search committee chairman Shapiro wrote to Vice President Mondale in June. In the letter, released yesterday, he said "the heart of the bureau problem traces back to the fact that over the years Mr. Hoover was not in fact accountable to the various attorneys general to whom he theoretically reported."

Bell acknowledged yesterday that the FBI is currently beset by morale problems. "They don't know what the law is, They don't have a charter. They're worried about the indictment [of an FBI official in a break-in case], and about the civil suits that are breaking out," he said.

The Attorney General said he "vacillated" over picking someone from inside or outside the bureau to head the organization at this crucial time before settling on Johnson.

Johnson was born in Winston County Ala., which declared itself on the side of the Union during the Civil War. His wife, Ruth, teaches in a largely black, public junior high school in Montgomery.