Sen. Harry F. Byrd (I-Va.) had never heard of Jane Cathcart before that day last year when the 49-year-old Warrenton woman drove to Capitol Hill, went to the politician's office door and told him a sob story of how she had been bilked by a homebuilder in 1977.

Yesterday that visit paid off.

For the first time in his 15-year career in Washington, Byrd inttroduced a bill that would benefit one person and one person alone: Jane Catheart.

Approved by the Senate Finance Committee yesterday, the piece of legislation will provide a one-time tax benefit of $10,000 to Cathcart, a divorced mother of four who was caught in a capital gains tax squeeze as a result of the 1977 incident.

At a hearing on Capitol Hill yesterday, Donald Lubick, an assistant secretary of the Treasury (which had opposed the bill) said, "if Senator Byrd is willing to use 15 years of chips on this one," the Carter administration would not object.

Cathcart's saga began in 1977 when she sold her Virginia home, purchased a piece of land in Warrenton and hired a builder, John A. Giggs, to construct a new house. A short time later, according to Byrd's office, Giggs fled to Florida with Cathcart's money, leaving a half-finished shell on the Warrenton lot.

Cathcart sued the builder and won a judgement of $100,000. But her troubles weren't over. Because of all the delays, Cathcart was unable to move into her new home in time to avoid paying capital-gains taxes from the sale of her old home. Under the law, a homeowner can avoid paying the taxes on profits from the sale of his or her principal home if the money is reinvested within 18 months in another principal residence costing at least as much. reinvesting, Cathcart also would have had to occupy her new home within two years of the sale of her old one.

Byrd proposed that Cathcart be given an extra three years to move, thus allowing her to avoid paying about $10,000 in taxes on the profits from the sale of her original home.

A spokesman in Byrd's office said yesterday that the bill, which still must be passed by the House and Senate, stipulates half a dozen criteria that specifically apply to Cathcart's case, though the legislation does not mention her by name.

Chances are slim that the bill would benefit anyone else, a spokesman in Byrd's office said yesterday. "It would be awfully hard for anyone to fall into the same category," according to the spokesman.

Because the Constitution requires that all tax bills originate in the House, the Finance Committee attached Jane Cathcart's tax break to a House-passed bill that would allow Ohio Wesleyan University to import a carillon without having to pay import duties.

In a telephone interview last night, Cathcart said she was "extremely grateful" to Sen. Byrd and "I could not have been treated more warmly." She said she thought it would be better "if the whole law were changed. Others in my situation should be able to benefit also."