In short frivolous rebukes means scolding in a silly manner.
Explanation:
This expression 'Frivolous Rebuke' in the chapter, The Portrait of Lady means the grandmother would scold the sparrows lovingly when they sat on her shoulders or at times head. After the grandmother moved to the city, life became a bit lonely for her. The author became busy in his education. She would spend the maximum of her time at the spinning-wheel. The happiest half-hour day for her used to be in the afternoon when fed the sparrows. She sat in the verandah to feed the sparrows. She broke the bread into little bits and threw them to the birds. Hundreds of sparrows collected around her chirping incessantly. A few of the birds sat on her legs, others on her shoulders. Some even sat on her head. During this time the grandmother used to talk to the sparrows lovingly.