。
She became deeply uneasy about her son's health.
Literal
She [topic-は] son [genitive-の] health [subject-が] very anxious became.
不安 ('uneasy, anxious') is a na-adjective that takes its source-of-anxiety with が — the source-of-anxiety + (が or を)になる pattern. Note that 健康 here is technically the が-marked subject — Japanese marks the thing one feels anxious about as the subject of the anxiety. Compare 心配する (with を), which more transparently treats the worry-source as direct object. 不安になる ('become anxious') is the change-of-state version: 不安だ ('be anxious') → 不安になる ('become anxious').