。
She made an effort not to cry.
Literal
She [topic-は] cry-not [so-as-to-ように] made-effort.
~ないように is the standard 'so as not to / in order not to' construction, attaching to the negative form of the verb. Pair it with 努める ('to make efforts toward') and you get the explicit, slightly formal phrasing 'made an effort to not cry.' 努める carries a more deliberate, almost officialese tone than 頑張る ('try hard') — it shows up in business contexts (努力する is its noun form) and reads as conscious, controlled effort. The everyday partner is the colloquial ~ないようにする ('try not to / make sure not to').