“I am really happy with whatever anyone calls me,” Mr. O’Rourke said in an interview. He said Texas voters were far more interested in discussing jobs, health care and immigration.

“They want us to take on these really urgent priorities,” he said, “not just this smallness and pettiness that dominates the national conversation today.”

But as Mr. Trump showed, derisive nicknames can plant doubt, as it did in the case of “Crooked” Hillary Clinton, and “Little Marco” Rubio during the campaign, and with “Low Energy” Jeb Bush, “Liddle Bob Corker” and “Cryin’ Chuck Schumer,” among others, as president.

Of course Mr. Cruz comes to this game with his own label liability: He was born Rafael Edward Cruz, but goes by “Ted.”

“Cruz lacks Trump’s visceral instincts,” said Paul Begala, a Democratic strategist. “Plus, I suspect voters are getting tired of Trump’s schtick from Trump; they’re unlikely to want a pale imitation.”

So if Mr. O’Rourke is going for a Latin sound to appeal to one segment of Texas voters, Mr. Cruz could be accused of veering Anglo-Saxon to appeal to another.

That Mr. Cruz would move so quickly to try to skewer his opponent is a measure of how seriously he is taking the challenge from Mr. O’Rourke, even though no Democrat has won a statewide office since 1994.