My wife is a diamond nerd, as well as one of the most rational people you'll ever meet, and she'd explain it this way: the diamond is a test that serves various useful purposes from the perspective of the woman:
1) It demonstrates the man's ability to provide; 2) It demonstrates the man's willingness to forgo the other things he could buy with the money for a gift for her that does not directly benefit him in any way; and 3) It demonstrates to the woman's peers her husband's social standing.
An important thing to remember is that women take on a huge opportunity cost in getting married and having kids. My wife and I just had our first baby. Pregnancy and childbirth is a barbaric process and a woman will never be quite the same after. Taking care of an infant is tiring, stressful, boring, and thankless--and no matter how enlightened the husband is and how willing he is to help out, because of the baby's hard-wired inclinations the buck will always stop with the wife. Finally, if the woman is educated and has a career, that career will be permanently compromised by virtue of getting married and having kids. With all this in the background, it is evident that the social convention of the ring conveys valuable information to a woman considering getting married and bearing these costs. Re: 1, all else being equal a man that can provide more is more valuable than a man that can provide less. Re: 2, all else being equal a man that is willing to forgo things for himself is better than a man who is not; Re: 3, all else being equal a woman's social life will be positively affected in a tangible way by marring a man of higher social standing than of a lower one. It is totally rational for a woman to ask: "I'm willing to give up all this, what are you willing to do?" The ring is just a way to demonstrate this commitment in a way that is universally understood and spares everyone the awkward conversation of laying all the chips on the table.
Now, these elements are less important in 2013 than they were in say 1813. But at the same time to an extent they are not diminished in importance, but rather culture has shifted to downplay the uncomfortable truths at the root of these elements. For example, we like to downplay how much women really give up by being the child-bearers. But for an educated woman, if having kids cuts the probability of attaining a high position on her career by half,[1] the choice to have kids could mean millions of dollars in lost income over a lifetime. It's totally rational for a woman thinking of making such a compromise to demand from her potential husband a gesture showing that he is willing and able to take the other half of the bargain. But all this is something that we don't say in polite company, because we have fully bought into the mythologies about marriage that are in vogue at the moment: that it's just about "love" and all that matters is a mate that makes you "happy" and that it's an arrangement that is costless to enter into and exit out of.
There is, of course, nothing inherent about diamonds that makes them the only option for this kind of signaling. Indeed, in other cultures a cash gift is the norm instead. But the value of a signal is in its standardization, so to speak, and it just so happens that the diamond is the standard signal in western culture.
[1] Less than 20% of women never have kids, but about half of very high achieving women (executives, etc) never have kids.