There are many versions of the story about how nonsense mutations were named, but this one is told by people who were actually involved in the first isolation and naming (and it is a good story, too).
Phage T4 was a favorite genetic model for the "phage group", a collection of scientists who played a pivitol role in the elucidation of the central dogma and the development of molecular biology. Nonsense mutants were first isolated in phage T4 by a group of graduate students at Caltech (who later became professors). The experiments arose over an argument between Dick Epstein and Charley Steinberg about whether it was possible to isolate a class of mutants that have a phenotype in one host but not in another host (at the time, other classes of conditional mutations had not yet been isolated). The problem was that to find this class of mutants required screening a large number of T4 placks on two E. coli strains, which is a lot of work. The following is loosly paraphrased from a letter Epstein wrote to Frank Stahl.
Charley agreed to help look for the mutants, picking 2000 plaques in the first try. We also managed to convince Harris Bernstein (then a graduate student working on Neurospora genetics) to help and offered him the dubious reward of naming the mutants after him. Harris had the nickname Immer Wieder Bernstein ("Forever Amber" in German). That night we isolated several of the desired mutants and named them "amber mutants".
Wild-type phage T4 forms plaques on both E. coli strains tested, strain CR63 and strain B. The T4(Am) mutants formed plaques on strain CR63 but not on strain B. They serendipitously found these mutants because strain CR63 had a preexisting amber suppressor mutation and strain B lacked an amber suppressor. Later others discovered the two other classes of nonsense mutations and named them ochre and opal. In addition to their use as conditional mutations, nonsense mutations played an important role in the elucidation of the genetic code and the properties of genes. The isolation of nonsense mutations also led to the idea that it should be possible to obtain temperature sensitive mutants, and subsequently phage T4(Ts) mutants were isolated by another Caltech student, Bob Egar.
As with every story that has been told many times, there are other popular versions of this story. According to Epstein, Amber mutations were named after Harris Bernstein as described above. According to Edgar, Amber mutations were named after Harris Bernstein's mother. However, both stories invoke the English translation of Bernstein as the rationale for the name.