The dedicated budget-cutters at the White House and in Congress are having second thoughts about one of their good works. It seems that as they sliced their way through the budget they acted to reduce the already small Social Security benefits of several thousand elderly nuns and other members of religious orders.

The affected orders have complained, and repentance is nigh.

The offending vote was to eliminate the $122-a-month minimum benefit under Social Security. This now serves to sweeten the monthly checks of some 3 million beneficiaries, whose low earnings records are such that they would otherwise be entitled to less. President Reagan said when he recommended wiping out the minimum -- which both houses of Congress have agreed to do -- that those whose imcomes fell too far could turn to welfare.

But for various reasons that apparently wouldn't do for the nuns and others involved, and now a group representing Roman Catholic nuns is asking Congress for an exemption from the proposed new rule.

The nuns' proposal was outlined last week to the Senate Finance Committee by Sister Frances Mlocek of the Leadership Conference of Women Religious of the U.S.A.

It would allow thousands of nuns and male members of religious orders who have taken the vow of poverty to remain eligible for the minimum benefit for at least 10 years; they would be differently treated from the others now on the minimum, even though many of them are also poor and elderly.

Mlocek said the nuns or male clerics benefiting from the existing minimum benefit provision constitute a special case. Until 1972, persons in religious orders who had taken vows of poverty were not eligible for Social Security. Only in 1972 did Congress permit them to enter.

The sister and 240 religious orders of men or women, with about 75,000 members in all, had enrolled. About 14,500 of the members are now on the rolls and many more will eventually become eligible as they reach retirement age.

Since they do not receive any cash pay, for tax and benefit purposes these individuals are deemed to have an income equal to the value of food, clothing and shelter given them by their orders. This is usually calculated at $100 a month or a bit more, meaning that the tax is very low and so the benefit would be too except for the minimum.

Mlocek said that since they were only made eligible for Social Security in 1972, the nuns and others involved have not had time to build up any earnings record sufficient to obtain a reasonable benefit. This argues for a special exemption, she said in an interview.

Rep. Bill Archer (Tex.), senior Republican on the House and Means Social Security subcommittee, said he thinks the nuns have a "justifiable cause." He said that it is his understanding that because of their vows of poverty, they can't apply for welfare.

An effort to grant an exemption, possibly in conference on the budget bill, but more likely in separate legislation, is picking up steam. Ways and Means Committee member James Shannon (D-Mass.) has drafted a bill.

Senate Finance Committee Chairman Robert J. Dole (R-Kan.) and Senate Social Security subcommittee Chairman William Armstrong (R-Colo.) have also expressed concern, and Sen. Daniel Patrick Moynihan (D-N.Y.) is expected to take up the cause.

"It's for the nuns? I'm for it," said Ways and Means member Thomas J. Downey (D-N.Y.). And in the administration there is reportedly similar sentiment.