What changes Lady Bracknell's mind and causes her to approve of Algernon marrying Cecily?

She finds out that Cecily has a lot of money and investments.

She sees Cecily and Algernon's deep love for each other.

She observes how smart and caring Cecily is.

Lady Bracknell approves of Algernon marrying Cecily after she learns that Cecily is actually very wealthy and has significant investments. This financial security changes Lady Bracknell's mind as she sees Cecily as a suitable match for Algernon due to her wealth and status.