Before getting into this trope, some context is needed:
The Philosopher's stone is THE stone. It's legendary alchemical substance capable of turning base metals such as mercury into gold or silver. Now this was before Christians decided that alchemy was witchcraft so bare with me. It is also called the elixir of life, useful for rejuvenation and for achieving immortality; for many centuries, it was the most sought goal in alchemy. The philosopher's stone was the central symbol of the mystical terminology of alchemy, symbolizing perfection at its finest, enlightenment, and heavenly bliss. Efforts to discover the philosopher's stone were known as the Magnum Opus ("Great Work"). The concept isn't strictly regional either, the idea of it is actually common in both Europe and some parts of Asia.
But how is it a trope? Well look at any media that has people looking for a red rock ( yes red, it's almost always red or a shade of red.), and possibly ignoring any and all warning about said rock.
Look at Fullmetal Alchemist, the Philosopher's Stone seemingly has the power to violate the law of Equivalent Exchange in alchemy and allow the user to perform miraculous feats. It turns out that the law isn't actually broken, but the stone is composed of human souls that serve as a massive battery.
Naturally enough, in Harry Potter and the Philosopher's Stone, said stone shows its traditional powers of turning lead to gold and creating the elixir of life. There is also a nod to its origins in that the Mirror of Erised is used to protect it by ensuring that only someone who wanted to find the stone but not use it is able to retrieve it. At the end, it's destroyed to prevent Voldemort using it, with Nicolas Flamel and Perenelle accepting death as "the next great adventure" after more than six centuries of life.
Does the stone have to be used for it's traditional uses? No. In Aladdin (the series not the movies), the stone was the source of all magic.
Does it have to be a stone? Not necessarily. When looking for a MacGuffin ( basically what both the good guys and bad guys are going after and will be explained in detail in another chapter), it is not always a physical object. So the stone could very well be a person. It could be a book, a plant or even water.
But there is always a catch.In the world where magic is real, there is a cost. You want a spell that can help with your crops, sure...but I hope you're okay with causing a drought else where. Same goes for the Philosopher's stone. You want immortality, well in order to make this extremely rare artifact you must sacrifice something. Your left arm, your sanity, your entire body, hell going for the extreme someone else's life. Plus...will it be worth it?
Say the stone is in fact a person. Are you willing to kill them in order to gain the power of the stone? What's to say that once you gain all their powers, that someone who also wants the stone will repeat the process?
Think people.When writing in a Philosopher's Stone, it's easy to forget people have emotions. Like the main protagonist wants the stone to heal their village, but slowly becomes more selfish as they reach their goal.Or give an exponential crisis: The pros and cons of using the power of the stone (the write ever so nicely wrote in that the stone had one use every X amount of years) to revive a friend or the love of your life. Hell make it more than one, but no more than 5 stones.
Just saying,a rare magic stone, will cause some bloodshed.