It's not much of a tradition at all. De Beers started pushing the use of diamonds on engagment rings in the early 20th century as an attempt to sell more diamonds. A tradition that the average American thinks has always existed is rather new and was artificially created by one of the most despicable companies to have existed[1]. I do credit them for creating one of the most brilliant marketing schemes of the past 200 years, but personally, I find it lamentable that they did for our society's sake.
From http://en.wikipedia.org/wiki/Engagement_ring
> Before the 20th century, other types of betrothal gifts were common. Near the end of the 19th century, it was typical for the bride-to-be to receive a sewing thimble rather than an engagement ring.[1] This practice was particularly common among religious groups that shunned jewelry (plain people). Engagement rings did not become standard in the West until the end of the 19th century, and diamond rings did not become common until in the 1930s in the United States, as a result of an extensive nationwide marketing campaign by the diamond industry. The phenomenon arose even later in other countries.[1] Now, 80% of American women are offered a diamond ring to signify engagement.[1]
> The idea that a man should spend a significant fraction of his annual income for an engagement ring originated de novo from De Beers marketing materials in the early 20th century, in an effort to increase the sale of diamonds.[1] In the 1930s, they suggested that a man should spend the equivalent of one month's income in the engagement ring; later they suggested that he should spend two months' income on it.[1] In 2007, the average cost of an engagement ring in USA as reported by the industry was US$2,100.[9]
[1] http://en.wikipedia.org/wiki/De_Beers#Controversies