There's a distinction in the USA (and I'm sure that it's a distinction in most places, but I'm not qualified to speak about those places) between "old money" and "new money". Typically, old money is inheritance, family stocks, trust funds, and general "dynastic wealth". New money is first-generation; it was gotten, through whatever means, entirely by that person instead of being gifted to them. Sometimes these can overlap, of course. Examples: A very rich man was born into a family of oil tycoons. He was a millionaire from the day he was born. He's "old money". A woman comes from a very impoverished background. Most of her family never even got to college. She becomes a wealthy investment banker through hard work, education, and a bit of luck. She's "new money". A man is descended from a family of British Lords. His family has been filthy rich since the 1700s. However, he also went on to found his own company and make substantially MORE money than he was ever even by his parents. He's a combination of "old money" and "new money". So the paragraph here is saying that Dallas was known for new entrepreneurs and recently-rich people. Their business savvy and whatever else allowed them to accrue wealth without relying on blood. And all of those old cotton barons simply couldn't compete with the new millionaires either in wealth or numbers, and so the "new money" was more powerful than the "old money".

There is a very well known French phrase 'nouveau riche' that is almost certainly the source of the phrase 'new money' - it means Newly Wealthy. It refers to the bourgeois people, merchants who had suddenly acquired wealth, even more wealth the the aristocracy had - the Aristocracy were the vieux(?) riche, the 'Old Wealthy'。 Such a phrase may go back many centuries, back to the ancient Greek Empire. 老鬼

"New money" refers to wealthy people who earned their money recently through business. "Old money" means wealthy families who have been wealthy for many generations. "New money" families are sometimes looked down on as less sophisticated in some circles.