。
She went by the name of Beth.
Literal
She [topic-は] Beth [quotative-という] name [with-で] was-known.
通る ('to pass, go through') extends idiomatically to 'be known by' — 名前で通る ('be known by [a name]'). The image is of a name 'passing through' as one's social label. ベス transcribes 'Beth' (or Bess); the trailing ス approximates the English /θ/ or /s/ — Japanese phonology lacks /θ/, so it standardly maps to /s/. ~という ('called, named') chains a label to a noun. ~ていた marks the past habitual/durative state.