Doctors at the Washington Hospital Center last night began an operation to give a kidney to a 2-year-old child whose life would be measured in months without it.

The kidney, which was donated by the relatives of a young car accident victim in North Carolina and selected as a suitable match by an intricate computer network, was packed in ice in a styrofoam box and flown to National Airport early yesterday.

While the operation on the child, whose first name is Elbert, marks the end of months of hope and desperation on the part of his foster parents and his doctors, it is not necessarily the final chapter.

There is a "50-50 chance" that Elbert's body will not accept the donated organ, according to Dr. Charles B. Currier Jr., who performed the operation and is the director of transplantation in a joint program involving both the Hospital Center and the George-town Univeristy Medical Center. In that care, the search for another kidney will have to begin again.

The next two or three months will be "critical" to determine whether the child's body will reject the organ. Currier said. But the operation itself is difficult in a child as young as Elbert, said Currier, who has performed 35 such tranplants under the two hospitals' program.

For Elbert, the drama heightened early this month, when both of his natural parents proved incapable to give their son a kidney. Since the chances of success of transplanting a kidney from a non-related living donor are no greater than with a kidney from someone who has died recently, doctors only considered the use of a cadaver kidney. Thus, Elbert's only chance for survival depended upon a stranger's death and the donation of his kidney.

Elbert's name was placed in a nationwide computer network that automatically matches his blood and tissue types with kidneys that become available. As a young child who could not survive long on the machine that acts as an artificial kidney, Elbert was given a high medical priority for the next suitable kidney. On Wednesday night, it became available.

Washington Hospital Center transplant coordinator Denis O'Donnell was gazing at a recently arrived serving of veal parmesan at a local restaurant when a call came from his counterpart at Georgetown Medical Center, Geordie Thomson.

Thomson had fed Elbert's name to the United Network for Organ Sharing computer after his mother and father proved incapable of donating a kidney. When the call came from a North Carolina hospital that a kidney matching Elbert's medical characteristice was available, the case became O'Donnell's responsibility. Washington Hospital Center performs transplants of cadaver kidneys while transplants from living donors are perfomed at Georgetown.

At 1:40 a.m. yesterday O'Donnell stood on a runway at National Airport to meet a chartered plane carrying the kidney. After delivering the kidney to a tissue typing laboratory, where further tests determining its suitability could be carried out, O'Donnell worked through the night to prepare for the operation.

By 7 a.m., the laboratory notified that the kidney had passed the last tests. O'Donnell called Currier and the other doctors involved in the case to tell them the operation could take place. "It sounds sort of cornball," O'Donnell said, "but the teamwork is really kind of inspiring. You don't usually see this kind of cooperation between hospitals."

Meanwhile, another phone call had been placed to Elbert's hospital room at Georgetown, where Irean Nelson, the child's foster mother, spent the night waiting for test results. "The doctor answered the phone and then he looked down at the crib," said Nelson. "He said, 'Congratulations Elbert, you've got a kidney.'"

Ninety other people waiting for kidneys in the Washington area, Currier said, and another 13,000 across the country. That is why he urges people to sign a card that donates their organs in the event of their death. "That way," he said, "at least some good can be reaped from an otherwise totally tragic situation."