Tim Agazio has the coolest name origin (even if it is just speculative). By way of comparison, my surname is an old English place name meaning "homestead by the hill," "estate on a hill," or "settlement on a hill." Tim's ancestors were named for holy warriors, mine for prime real estate.
The spelling is usually "Dunham," though the "Donham" spelling was sometimes used in the 18th century when my forebears lived in Plymouth, Massachusetts, and adjacent towns. When my ancestor, Moses Dunham, moved to Hartford, Maine, he took with him the "Dunham" spelling. When his brothers Thomas and James Thomas migrated to the nearby town of Hebron, they were denominated "Donham." It's hard to tell whether this was a deliberate choice (doubtful stories abound of men who changed the spelling of their surnames to distinguish themselves from siblings). Bill West of West in New England descends from a son of James Thomas Donham who readopted the "Dunham" spelling.
The Dunham name is neither rare nor common in the U.S. It became more common when a shady character named Jonathan Singletary began using it as an alias. Before absconding to New Jersey, he raised some hell in Plymouth.
Wheras Jonathan Dunham, alias Shingleterry, hath long absented himselfe from his wife and family, tho advised and warned by authoritie to repaire to them, and for some considerable time hath bine wandering about from place to place as a vagabond in this collonie, alsoe deseminating his corrupt principles, and drawing away another mans wife, following him vp and down against her husbands consent; and att last hee meeting with and accompanying a younge woman called Mary Rosse, led by inthewiasticall power, hee said hee must doo whatt shee bad him, and according did, both of them, on her motion, att the house of John Irish, att Little Compton, kill his dogg, against the declared will of the said Irish; and although hee put them out of his house, yett they would goe in againe; and according to theire antick trickes and foolish powers, made a fier in the said house, and threw the dogg vpon it, and shott of a gun seuerall times, and burnt some other thinges in the house, to the hazard of burning of his house and younge children, keeping the dores and not opening them to the said John Irish when he come with some of his naighbors to rescue the same; to the disturbance of his maties peace commaunded, and against the laws. [Records of the Colony of New Plymouth in New England (1856), 6:113f]He was sentenced to be "publickly whipt att the post" and was booted from the colony. Among his descendants was Ann Dunham, the mother of Barack Obama.
The most famous person to bear my surname was a remarkable woman named Katherine Dunham. A pioneer of modern dance, she was the daughter of Albert M. Dunham, a dry-cleaner in Joliet, Illinois, and the granddaughter of John Dunham, a former slave who settled in Memphis after emancipation. If our paternal lines intersected, it was probably in some millennium long past. Katherine died last year, but her legacy lives on at the Katherine Dunham Centers for Arts and Humanities.
I should also mention Dunham Bootmakers, a company founded by three brothers from North Paris, Maine—just a stone's throw from my hometown of Greenwood. They were distant cousins of mine, but neglected to mention me in their wills.