。
She was dying to see her child.
Literal
She [topic-は] child [target-に] want-to-meet, couldn't-bear.
~たくてたまらない is the emphatic frame for urgent, unbearable longing — 'dying to do X, can't stand not to do X.' It splices the desiderative ~たい with ~てたまらない ('cannot endure, unbearably'), connected through the te-form of an i-adjective (~たい → ~たくて). The verb 会う takes に, not を, in standard Japanese: you 'meet up with' (に) someone rather than 'meet' (を) them. The whole construction carries strong emotional weight and is common in scenes of separation or longing.