In From the Mixed Up Files of Mrs. Basil E. Frankweiler, what did Mrs. Frankweiler admire about Claudia?
It seems clear that when Mrs. Frankweiler finally meets Claudia and her brother in the story, she is impressed by Claudia's maturity and the way in which she remains fixated on proving the truth about Angel and Michaelangelo, even when Mrs. Frankweiler herself tries to get them to think about going home to their parents. In addition, I think Mrs. Frankweiler, from her plan of wanting to "help her to see the value of her adventure," recognises that Claudia is such a determined and motivated young lady that she has not learnt what she needs to learn yet about herself and, in a sense, she needs to finish this quest before she can return to her parents.
It is these qualities that Mrs. Frankweiler notices about Claudia, and this causes her to give her and her brother a chance to find out what they want to know as a kind of puzzle. It almost seems as if Mrs. Frankweiler sees something of herself in Claudia, and realises that she can't patronise her or treat her as a child.
Mrs. Frankweiler admired Claudia's thirst for knowledge. She also admired Claudia for not giving up on a challenge and seeking out the truth. She felt that Claudia would always be able to keep her secret because it made Claudia feel special which is what she wanted most at that point in her life.