Fontane spent several years in England as correspondent for Prussian newspapers. One of the English-language authors he read was Walter Scott; he even translated a number of Scott's poems into German.
The name "Effi" was inspired by the name Euphemia or Effie Deans in Scott's novel The Heart of Mid-Lothian:
“The case of Effie (or Euphemia) Deans,” resumed Saddletree, “is one of those cases of murder presumptive, that is, a murder of the law’s inferring or construction, being derived from certain indicia or grounds of suspicion.”
“So that,” said the good woman, “unless poor Effie has communicated her situation, she’ll be hanged by the neck, if the bairn was still-born, or if it be alive at this moment?”
However, there are hardly any parallels between Effi Briest's and Euphemia Deans's stories, so Effi's name is probably not intended as an illusion (Schmidt, 1998).
"Briest" is the name of a noble family that had become extinct and which Fontane had come across during his research for Wanderungen durch die Mark Brandenburg. The name appeared in a section entitled "Das Pfuhlenland" in the first edition of Wanderungen ... that was later removed (Schmidt, 1998).
Briest is not the only name from an extinct family that Fontane used in Effi Briest. "Güldenklee" is another example. Names of extinct families had the advantage that they provided some local colour to the story without offending real persons who might feel they were represented or alluded to in an unfavourable way. (In fact, a certain von Glasenapp complained about this because he thought that the name "von Grasenapp" in the novel was an allusion to his family name. "Von Instetten", by contrast, is a name that apparently never existed. See Schmidt, 1998.)
Fontane was proud on his choice of the name "Effi Briest" because it sounded nice with all those "e"s and "i"s, as he wrote in a letter to Julius Rodenberg. For a long time, he had considered a different name, namely, Betty von Ottersund.
Based on this, it is doubtful that readers should seek a "meaning" behind the name.
- Schmidt, Michael: "»Geheimnisse […] und Anspielungen« oder Caroline und Effi von Briest. Namen-Anspielung‹ und Proto-Emanzipation in Theodor Fontanes Roman", Nordlit 3 (1998)