。
She doesn't give up over little things.
Literal
She [topic-は] [a-little-thing-ちょっとのこと] [for-では] doesn't-give-up.
ちょっとのこと is the noun-form counterpart of ちょっとの間 ('a little while') — 'a little matter, a small thing.' で here marks the cause/reason ('over X, on account of X'), and the contrastive は inserted after で gives ではあきらめない the flavor of 'as for [a little thing], (she) doesn't give up' — pushing back against an implicit 'maybe she would.' The combination paints a portrait of resilience: small obstacles aren't enough to stop her.