As an artist, this issue of "what gives something with no blatantly functional purpose value?" is a question that I have to frequently address. Clearly you're not all that taken with Diamonds, and the attitude you've displayed here leads me to guess that you're probably not all that interested objects of intangible value. That's fine, but as someone whose life is devoted to the study and creation of objects with intangible value (in my case paintings) I do think you're missing out.
Now diamonds and jewelry crafts are not my forte. That said, a well presented diamond is truly a beautiful thing. Forged from carbon at unimaginable pressures, a diamond of value has been cut by an expert stone-cutter. I don't know if you've ever tried your hand at stone-cutting, but it is not an easy task, and to craft a beautiful diamond of mathematical perfection from a roughly hewn chunk speaks to the hands of a talented artist. Likewise, setting a diamond is no simple task. As a stone whose beauty is optimized only in maximum light, a diamond must be set so as to allow a lot of light into the stone. This means the creation of delicate settings, shaped and soldered (generally out of sterling silver or a similarly rare and difficult to work with metal) by someone with the training and experience to properly handle such metals without destroying them [and they are VERY easy to destroy].
When you see a diamond like this, it is not the diamond itself which is beautiful, but instead it is the way that light refracts within the diamond. Thusly a diamond becomes a reference not just to the unimaginable forces which formed it, or the expert hands which prepared it for presentation, but also to the very light which is our primary method of understanding the world around us. To truly appreciate such an ephemeral object, one must examine it closely and in person. If you have never held a well prepared diamond in your hand, you have never actually seen a diamond. It is not the chunk of white stone which we normally think of a diamond as being, instead its properties are such that it presents all the light and colors in its immediate vicinity. To appreciate a diamond the way it was meant to be appreciated, one must appreciate the world in which it exists, as the diamond reflects and returns the world to the viewer. If it has value, it is in the fact that its beauty reminds us to examine the beauty of the world around us.