In one of the classes I’m currently taking we’re assigned the book “Lies My Teacher Told Me” by James W. Loewen. In the second chapter he states that “Ancient Roman and Carthaginian coins keep turning up all over the Americas, causing some archaeologists to conclude that Roman seafarers visited the Americas more than once.” I was wondering how much evidence supports this claim and if so where in the Americas did they land?
Short answer:
There is no compelling evidence for this suggestion.
Discussion:
Coins are durable, and not proof that the people who made them carried them to the location you might dig them up at a later date. Recall that the monetary value of these coins were their weight in silver or gold-- rulers might on occasion melt down old coins and restrike them-- but they didn't have to. An ounce of silver with Marcus Aurelius' head on it was still an ounce of silver.
Roman coins have been found in Okinawa, perhaps brought by the Portuguese or Spanish. African coins, from the Kilwa Sultanate, have been found in the Northern Territory of Australia.None of this is evidence that the people who made the coins visited the place where they were found.
We have no evidence that the Romans or Carthaginians crossed the Atlantic. Britain appears to be as far West as they got, and just what their contact in Ireland might have been remains controversial. There were clearly trading links, but how much more than that is uncertain.
To conclude that a Roman coin was brought by a Roman sailor or merchant requires much more than the coin itself, rather you'd need to know exactly where it was recovered from and the age of the material around it. When [if] we find Roman coins buried at Tikal, then there will be something to talk about; but Roman coins were and are quite common-- a Spanish or English sailor could easily have have had a Roman coin dug up from home. Similarly, we find Roman coins in medieval Italian graves . . . that doesn't make these Lombards "Romans" . . . just means they had Roman coins.
You can contrast this with India and Sri Lanka, where we can find Roman coins which are associated with good evidence of Roman trade, and references in sources like Tacitus and Pliny. The Romans leave us quite a number of descriptions of places they visited, none suggest the Americas. The Romans have a name for Lanka -- they call it "Taprobane" which fits the Sanksrit name "Tamraparni".
So if you're in search of visits by Romans to faraway shores, the Indian Ocean is a better bet than the Atlantic. We can also observe that the Romans weren't enthusiastic blue water navigators. They hugged shorelines when they could, and we don't see much evidence that they would have ventured across open ocean like, say, the Polynesians. It's also notable that we don't find evidence that they got to the Azores, which is much closer.
Sources:
Bursche, Aleksander. “Circulation of Roman Coinage in Northern Europe in Late Antiquity.” Histoire & Mesure, vol. 17, no. 3/4, 2002, pp. 121–141.
Haselgrove, Colin, and Stefan Krmnicek. “The Archaeology of Money.” Annual Review of Anthropology, vol. 41, 2012, pp. 235–250.
CAHILL, JAMES. “TWO THIRD-CENTURY ROMAN COINS FROM MADDOCKSTOWN, CO. KILKENNY.” The Journal of the Royal Society of Antiquaries of Ireland, vol. 141, 2011, pp. 190–198.
TRAVAINI, LUCIA. “Saints and Sinners: Coins in Medieval Italian Graves.” The Numismatic Chronicle (1966-), vol. 164, 2004, pp. 159–181.
McLaughlin, Raoul. “Ancient Contacts: The Roman Emperor and the Sinhalese King.” Classics Ireland, vol. 21-22, 2014, pp. 1–40.
It's already been discussed that there is no evidence for this. Unfortunately, as I outline in this answer Loewen repeatedly pushes such bogus ideas in the name of a compelling story- evidence and reason be damned.