She was desperate to see her mother, even if just a glimpse.
Literal
She [topic-は] mother [object-を] one-look want-to-see unbearably.
~てたまらない ('unbearably, can't stand it') expresses intense, uncontrollable feeling. 一目見たい means 'wanting to see even once/briefly.' The combination paints a vivid picture of desperate longing.