Frayda Levin, a New Jersey libertarian activist and former small-business owner, is a woman of many passions: promoting liberty, ending marijuana prohibition and opposing her state’s recent minimum-wage increase. But Ms. Levin has added another cause as well. At gala benefits for free-market research institutes and at fund-raisers for antitax groups, she has urged like-minded donors to help send Senator Rand Paul, Republican of Kentucky, to the White House.

“I consider that one of my main goals,” said Ms. Levin, who has met with Mr. Paul several times and in February introduced him at a private conference in Florida hosted by the Club for Growth, a conservative advocacy group. “I tell people he’s the Republican of the future. He’s got both the intellectual heft and the emotional understanding.”

As he has risen in prominence as a Republican presidential contender, Mr. Paul is avidly courting a small but influential cluster of wealthy libertarians. His pursuit offers an intriguing window into an eclectic network of potential donors who have made fortunes in Silicon Valley start-ups and Wall Street hedge funds, a group that could form a vital donor base if he makes a bid for the Republican nomination. A tight-knit tribe of philanthropists and entrepreneurs, they have exerted enormous intellectual influence on conservative policy. But they have historically spent more on nonprofit groups and endowing college economics departments than they have on backing candidates.

Now Mr. Paul is seeking to nudge them into the comparatively messier world of presidential campaigning.