In his news conference yesterday, President Reagan used the story of an unnamed "little girl" who has been hospitalized most of her life to show how federal health regulations can "be a tremendous expense to the taxpayer" and yet "do no good to the patient."

He said that although her doctors and parents would like her to receive care at home, at a cost of $1,000 a month, Medicaid regulations require her to stay in the hospital at $6,000 a month or lose coverage.

The reference caught officials at the Department of Health and Human Services, which administers the regulations, by surprise and sent them scrambling, particularly since the answer came in response to a question about budget cuts which seemed to have nothing to do with the problem in question.

A White House spokesman said later that the child's problem had come to the attention of Vice President Bush through Rep. Thomas J. Tauke (R-Iowa).

Tauke's office provided the particulars:

The "little girl" is Katie Beckett of Cedar Rapids. She had been hospitalized for most of her 3 1/2 years, following complications of viral encephalitis, an inflammation of the brain.

Because of a "Catch 22" in the law, Katie can receive government assistance while in the hospital but would lose coverage if she received care at home because her parents' income is above eligibility levels.

A case worker in Tauke's office said she was recently informed by HHS that there are "no exemptions."

But yesterday, when the case drew presidential remark, HHS officials sought for several hours to figure out what he was talking about. One spokesman suggested that the problem had already been taken care of at the federal level in new legislation that would allow states to get waivers for home health care.

Later, however, Paul Simmons, deputy commissioner of the Social Security Administration, called to say that there was still a problem in terms of eligibility.