。
She objects just for the sake of arguing.
Literal
She [topic-は] argument [for-the-sake-of-のために] protests.
~のために is the standard 'for the sake of / in order to' construction, attaching to a noun with の in between. The exact reading is context-driven: 子供のために ('for the children') = beneficiary, 健康のために ('for health') = goal. Here 議論のために gives a flavor of *purpose* — protesting in order to generate an argument, not because she has a position to defend. The character sketch is sharp: someone who lives for the dispute itself. 抗議する is a firmer 'protest, object,' more weighty than 反対する ('oppose') or 文句を言う ('complain').