Her eyes lit up at a single hint, and the girl set her cute cat-shaped mechanical pencil flying across the notebook.
Literal
One hint [to-に] eye [obj-を] lit-up girl [topic-は], cute [attr-な] cat-shape [gen-の] mechanical-pencil [obj-を] notebook [on-に] sets-running-[goes-ていく].
目を輝かせる ('to make one's eyes shine / to light up with excitement') is a fixed causative-form idiom: the transitive form of 輝く ('to shine') expressing 'to cause one's eyes to shine.' ひとつの ('one, a single') emphasizes the small trigger. ~させた is past causative, used attributively as a relative clause: 'the girl who made her eyes shine.' ペンを走らせる ('to make a pen run / to write energetically') is another causative idiom — 走る ('to run') + ~せる, for the metaphor of quick, flowing writing. The stacked causatives are a hallmark of descriptive prose.