She ate some cookies to tide over her hunger until dinner.

Literal

She [topic-は] dinner [until-まで] hunger [object-を] endure [purpose-ため] some cookies [object-を] ate.

しのぐ (凌ぐ, 'to endure, weather') is used for getting through a difficult period. ため marks purpose. いくらか ('some') is a quantity expression.