For a race that is supposed to have national significance as a test of public reaction to "Reaganomics," the New Jersey governorship campaign is attracting relatively little attention in the place that counts most -- New Jersey.

A poll released last week by the Eagleton Institute of Politics at Rutgers University found only half the registered voters sampled could, without prompting, name the candidates -- former assembly speaker Thomas H. Kean (R) and Rep. James J. Florio (D). A debate between the candidates Thursday night got far-from-banner coverage in the papers.

"It's like shouting down a well," said a media adviser to one of the candidates. "It just gets swallowed up."

Part of the problem is the New Jersey predicament of having no VHF television stations of its own, which means most new gets filtered through Philadelphia or New York; the debate Thursday was carried only on a UHF public broadcasting station. Also, both candidates have accepted public financing, with a spending limit of $2.1 million, so they are hoarding their advertising money for the closing drive. And the absence of an incumbent in the race -- Gov. Brendan T. Byrne (D) must step down after two terms -- makes it harder for the voters to focus.

Still, the race is legitimately of interest outside the Garden State. New Jersey and Virginia are the only states electing governors this year, and the results will be interpreted willy-nilly as a measure of the health of the Republic Party one year after Ronald Reagan's election.

There is a lot of disagreement about how fair that test will be in the case of New Jersey. On paper, Florio ought to be a strong favorite. The Eagleton Poll had him 8 points up but noted that half the supporters of both men said they might change their minds before Election Day.

Democrats have controlled the governor's office in Trenton for 24 of the last 28 years and have a big registration edge. Florio, 44, who was elected to the House in 1974 has built a strong legislative record as a defender of railroad passenger service and an advocate of stiff cleanup measures for toxic waste dumps -- both issues with bipartisan appeal.

He comes from south Jersey, an area increasingly sensitive to its lack of influence in state government. Advisory referenda on "secession" have carried in several of the southern counties. In the June primary, Florio was able to create such a solid block vote in the eight southern counties that he defeated 12 northern candidates.

A strategist in his campaign likens his situation to that of Jimmy Carter in the 1976 presidential contest: "If Jim can get south Jersey to vote for him as a southerner and the north Jersey cities to vote for him as a Democrat, there's no way he can be beat."

But Kean has solid credentials of his own, and a "time for a change" theme that shows up in all the polls and worries Florio. Byrne is going out of office as an unpopular governor; the recent naming of the Meadowlands sports arena for him is viewed universally as a symbol of his political insensitivity. In the debate Thursday night, Florio was walking a narrow line -- defending the Democratic record while asserting that his tyle of governing would not resemble Byrne's.

Kean has made the economic situation of the state the focus of his campaign. The wealthy scion of a family long prominent in state politics, Kean, 46, built a record as a political moderate in the legislature. He was runner-up four years ago in the GOP gubernatorial primary.

Seeking to separate himself from the pack in this year's eight-man primary, Kean put all his emphasis on a tax-cutting plan that he said would complement Reagan's national strategy and enable the state, which lost 80,000 manufacturing jobs in the last eight years, to compete better for industry.

Kean's plan, which helped him win the primary, calls for a complete phaseout of the corporate net worth tax, a reduction of the corporate income tax and an eventual reduction in the sales tax.

Florio calls the Kean plan "unrealistic," and says it would force the same kind of "severe" slashes in state services that Reagan's tax cuts have brought in federal aid. Kean, in turn, has accused Florio of perpetuating "big government" of the Carter-Byrne model and has pressed him to pledge that he will not increase state taxes -- a pledge Florio so far has refused to make.

Private polls show that in New Jersey, as elsewhere, the very broad initial support for Reagan's economic program is polarizing on partisan lines -- a threat to Kean. "We'd like to bring the campaign back to state issues," sayd Roger Bodman, Kean's campaign manager.

To that end, Kean has publicly dissented from administration policies on Social Security cuts, the environment, mass-transit subsidies and the sale of Aircraft Warning and Control System planes to Saudi Arabia, and has supported the Equal Rights Amendment.

But his basic strategy is still to split the dominant blue-collar vote by an anit-spending, anti-taxing stance, as Reagan did in carrying New Jersey by almost 400,000 votes a year ago. Eager to consolidate that victory and gain leverage on the redistricting of the 15 House seats, the national GOP is contributing almost half of a $2.1 million fund the state GOP will spend to supplement Kean's own campaign. (Democrats, with only $49,000 from Washington, will spend only two-thirds as much to help Florio.)

Reagan himself is scheduled for a mid-October visit; Vice President Bush has been in twice and is coming back once more, and there will be a parade of Cabinet officials, led by Secretary of Labor Raymond J. Donovan, who ran Reagan's New Jersey Campaign.

What all that will really prove about the acceptance of "Reaganomics" is hard to say -- especially if the Phillies are involved in World Series play until late October and no one in south Jersey is reminded that there is an election going on.