。
She suggested he try doing it.
Literal
She [topic-は] he [to-に] it [object-を] [why don't you try-てみたらどう] [quotative-と] said.
~てみたらどう ('how about trying ~?') is a soft suggestion pattern made of three parts: ~てみる (try doing), ~たら (conditional 'if'), and どう ('how'). The whole thing literally amounts to 'if you tried, how would it be?' but functions as 'why not give it a try?'. Wrapped in と + 言う for reported speech.