"In this book are the names of many Lowland Scots who migrated to Londonderry during the seventeenth century. These early settlers can be considered to be the founding fathers of the Scots-Irish in the region. One hundred years later, after the first colonies of British settlers were established in both Virginia and Londonderry, the descendants of Ulster planters began to emigrate in increasing numbers to the colonies of New England, Pennsylvania, Virginia and the Carolinas. Between 1718 and the War of Independence, half a million Scots-Irish left Ulster for the British colonies in North America. Many of their forbears can trace their origins to the early period of British settlement in county Londonderry. Surnames can provide much useful insight, if not proof, of family origins in Ireland..." [Source: http://www.ulsterheritage.com/forrest/londonderry.htm]
Bradford is including in this list of surnames. There is no proof that this is our Bradfords, but at least there is precedent for finding the name in Ulster. I have seen the Bradford name mentioned in a few other records of other Ulster counties.
While we know where Rebecca Bradford's Scots-Irish father was from from County Donegal, we still have almost no clue as to the origins of her maternal grandfather, Samuel Bradford of Red Lion Hundred, Delaware.
As I have written before, the one clue we have is that Red Lion Sam's son was buried in a Presbyterian church yard (Old Drawyer's Church in Odessa, Newcastle County, Delaware), and that was the predominant religion of the Scots-Irish. Just the fact that the family was living in that region of America is a clue, as many Scots-Irish settled there. Samuel's will gives no clue as to where he is from, nor do the land records wherein he is mentioned. Taking all this into consideration, plus the fact that there are very few records extant for the Scots-Irish in Ulster, finding Red Lion Sam's origins is going to be very difficult, but hopefully not impossible!