Rhodri Molwynog, also known as Rhodri the Bald and Grey, was an 8th century king of Gwynedd. Little is known about him, but he is called King of the Britons in the Annals of Wales, and he probably died in 754 A.D. Rhodri the Great (820-878) was also a Welsh king, named Rhodri ap Merfyn and also known as Rhodri Mawr, and he was from either the Isle of Mann or Manau. His father, Merfyn Frych, was King of Gwynedd until 844, who took the throne after the previous line (Cunedda) ran out of male heirs. In the Annals of Ulster he is called King of the Britons. Fun fact: it may be possible that rap superstar Eminem (Marshall Mathers) is descended from him. Then we have Rhodri ap Gruffudd (1230 - 1315), a Welsh prince third or fourth in line to inherit the throne. His uncle sent his father to London as a prisoner, where he died, in order to ensure his position as king, since Rhodri's father was Llywelyn the Great's oldest son, although illegitimate. When Rhodri's uncle died, his brothers fought for the throne, but Rhodri managed to stay out of it. Later, when all his brothers had passed, he would've been the only one with a legitimate claim to the throne of Gwynedd and the title Prince of Wales, but unfortunately he had sold his rights to his older brother. His son made no claim to the throne either.
There are a handful of other British and Welsh namesakes that are more modern. Rhodri Morgan may be the most important modern namesake. A Welsh politician, he was the First Minister of Wales.
In the U.S. this name is rare and unusual, with no record.