During their first meeting, which statement about the young woman's life provides Holmes with a good clue?

There are several points during Miss Stoner's introduction to her case that we might see as clues for Mr. Holmes. I'll list them off and you can choose which you think was most influential to the investigation.

First, Miss Stoner tells us that her stepfather is the last surviving member of one of the oldest and wealthiest families of England. Due to poor management of affairs, his family's fortune has been wasted over the past...

There are several points during Miss Stoner's introduction to her case that we might see as clues for Mr. Holmes. I'll list them off and you can choose which you think was most influential to the investigation.

First, Miss Stoner tells us that her stepfather is the last surviving member of one of the oldest and wealthiest families of England. Due to poor management of affairs, his family's fortune has been wasted over the past century. When Dr. Roylott married Miss Stoner's mother, he gained access to her wealth. Because her income was by inheritance of her family's wealth, Dr. Roylott would really only have control over the money in the event of her death. Helen mentions that her mother died under mysterious circumstances several years prior, and the money she left behind supported Helen, her sister, and their stepfather. In fact, they were so well off that Dr. Roylott stopped trying to set up a business in London and moved the family out to his country estate. 

From these statements, we might infer that Dr. Roylott married Helen's mother for her money. We come to assume that he was responsible for the mother's death, but even if he was not, he had much to gain from her death. Having used up all of the money his wife left behind-- save Helen and Julia's inheritance-- he might have been driven to kill the daughters in an effort to have access to their money.

Next, there is the mention that Dr. Roylott keeps many exotic pets. This might have tipped off Holmes as to the means of murder.

Finally, when Helen comes to describe her present circumstances of her recent engagement to be married and hearing whistling in the night, Holmes calls particular attention to her stepfather. He finds it suspicious that she would leave mention of him out of her story, especially as they live in the same house. Holmes discerns that he was quite angry with her, most likely about the engagement, as she has a bruised hand-print on her wrist. I think that if anything were to point Holmes' suspicions towards Dr. Roylott, it would be the fact that Julia Stoner died after becoming engaged, and now that Helen is engaged she fears for her life.