Clearly we need to know something about Hungarian adoption laws (which may supersede or be subservient to privacy laws). Hopefully others will contribute.
Working from the known to the unknown, and as Luke may have been alluding to in his comment, I assume your close relation knows the names of the adoptive parents (whether you do or not). What basis/knowledge does he or she have for knowing his/her birth date and birth location?
Does he/she know the names of the adoption lawyers/agencies in the US and/or in Hungary? If not, can he/she discover that information (whether or not for a fee)? If the Hungarian agency is discoverable, will they communicate with him/her?
Telephone directories were still quite popular in the US in 1991. Were these directories popular then in Hungary?
Given a surname and a year of birth/birth location for the mother, you then have some basis for extending a separate "fishing" line back in time. You can broadly search as to date/time, but specifically search as to the birth mother's maiden name and birth location. Privacy laws may restrict your access to common records, but again newspapers and directories might be accessible, all of which is quite location dependent.
Hope this gives you some ideas.
P.S. Were I in your shoes, I would frame out a plan and develop a base of information. Once I felt I'd exhausted reasonable access to Hungarian sources, I'd hire a Hungarian professional to advise me on the next steps to take.