"Youll Never Walk Alone", number one in UK in 1963 by Gerry and the Pacemakers, from the musical, Carousel or West Side Story, not sure which one, I will check and post answer.
The song has been recorded literally hundreds of times since the first chart entry by Frank Sinatra in 1945, so the version of this popular Graduation Song your mom remembers may be difficult to pin down. Pre-1955 versions include those by the aforementioned Mr Sinatra as well as Louis Armstrong and Perry Como.