Asserting that they have learned a lesson from their failure last year, House Democratic leaders are putting together a concentrated, computerized campaign to win early approval in the new Congress of public financing for House elections.

Supporters hope to bring the bill, which was given the number H.R. 1 to emphasize the leadership's interest in it, to the House floor quickly, while memories of the heavy-spending 1978 campaigns are still fresh. And the leaders say they will avoid the internal squabbles and extraneous issues that helped kill the proposal in the House last year.

Still, even the strongest backers of public financing give the bill no more than a 50-50 chance of passage in the House this year. Senate approval is even more uncertain, because conservatives there have promised a filibuster against any public financing bill -- even one that applies only to House campaigns.

H.R. 1 would offer federal matching grants, paid out of the fund established by the $1 income tax checkoff, to House candidates in general elections. A candidate could choose whether or not to accept the federal money. Those who did would get matching grants for each dollar of private contributions raised, up to $50,000 in federal money. But they would have to agree to spend no more than $195,000 -- from all sources -- on the campaign. Those who choose not to participate would have no spending limit.

In content, this plan is similar to the bill that was defeated last year. The differences this year are the atmosphere in which the bill will be considered and the strategy planned by the leadership to sell the legislation.

"What's new this year is the enormous inflation that hit campaign spending in 1978," says Bob Moss, counsel to the House Administration Committee, which has jurisdiction over the measure.

According to the Federal Election Commission, the average House candidate spent about $70,000 in the 1976 general election. Preliminary figures for 1978 indicate the average was more than $90,000 -- a 30 percent jump.

"Those numbers scare a lot of members," Moss says. "If we can get this bill on the floor while people still remember how much the challenger spent against them last fall, we might get a lot of them to say, 'Hey, this public financing would be better for me.'"

To help make that point, the House Administration Committee is preparing computer printouts on the spending patterns in individual members' campaigns to help in formulating details of the legislation.

"When we take a bill to the floor," said Rep. Frank Thompson (D-N.J.), the committee chairman, "we can use these to show people that our proposal would fit with the kind of spending they did in their last campaign."

In addition, the leadership is determined to avoid the bitter political fight that was provoked last year when Thompson added to the public financing bill a provision limiting the amount political parties could contribute to congressional condidates.Republicans saw this as a blatant gimmick designed to aid Democrats, and GOP leaders made the legislation a major party issue.

Thompson's move also caused a split among supporters of public financing. David Cohen and Fred Wertheimer of Common Cause, who lobbied hard for public financing, were sharply critical of Thompson's partycontribution limit.

Scars from that internal battle still sting. Although Wertheimer says he is perfectly willing to work with Thompson this year on public financing legislation, the feeling is not reciprocated. "Cohen and Wertheimer will not be welcome here," Thompson said recently. "They'll be welcome to testify, but I don't see how I can work with those guys beyond that."

But even if the idea's backers settle their differences and produce a bill free of controversial riders, the outlook for public financing legislation is dim in the 96th Congress.

Republicans, on the whole, still suspect a ploy to maintain the large Democratic majorities in Congress. Many entrenched Democrats, meanwhile, fear that public financing might be disastrous to senior members who lack the will or energy to fight a well-financed Republican challenge.