The girl who sat next to me [is or was?] named Stephanie.
In short, both phrasings are grammatical, and both hold the possibility of unwanted implicatures. One option for Becky is to explicitly cancel it. She could add a parenthetical statement like, “In fact, her name still is Stephanie.” Another option is to try to phrase things in such a way that the implicature never gets off the ground. For example, the grandfather could have said, “Then as now, they made the best milkshakes.” And Becky could write, “The girl who sat next to me was and is named Stephanie.” The trouble with these measures is that the extra words they require may impede the flow of the text and sound clunky. Only you can decide whether they do, and whether that is or isn’t the effect you want.
1. Comrie, Bernard. 1985. Tense. Cambridge University Press, pp. 41-42.