In The Adventures of the Speckled Band, why did Doyle use the untimley death of a lady to capture the readers attention so early on ?in the 2nd paragraph of the 1st page?
Doyle is writing a mystery; he wants to peak the reader's interest as soon as possible. The word you are asking about, "untimely", is more fully understood if you consider the phrase before it. Watson is about to reveal to the reader a long-held secret which could not be revealed until the lady whose promise he kept released him making his knowledge public. Here's the whole line:
It is possible that I might have placed them upon record before, but a promise of secrecy was made at the time, from which I have only been freed during the last month by the untimely death of the lady to whom the pledge was given.
Remember, too, that the original publication of the Holmes stories appeared in magazines, and keeping the reader's sometimes flighty attention (pulled by other visuals, like ads, though those were not as numerous as they are now) could be a challenge. Also, magazines often don't publish the story in sequential order, ie, it may begin on one page and be continued three pages later; again, keeping the reader "with you" is paramount in mystery-telling.
But above all, the reader wanted and expected a "can't put it down" mystery, which is exactly what Conan-Doyle delivered time and again.