There is so much information today on the web. I found, what appears to be some good history on both Thompson and Madigan names. Similar content was also located in The Dictionary of Irish Family Names by Ida Grehan.
Thompson
This famous name is one of the patronymic forms of the name Thom or Tom, diminutives of the male personal name Thomas. The given name is of Biblical origin, being an Aramaic byname meaning "twin", borne by one of Christ's disciples; in England the name Thomas was found only as the name of a priest before the Norman Conquest of 1066, but thereafter became one of the most popular male personal names, generating a wide variety of surnames. The patronymic forms from diminutives, such as Thomson (the Scottish form) and Thompson, found mainly in England and Northern Ireland, appear in the 14th Century, the first recording being from Scotland. The intrusive "p" of the English and Irish forms was for easier pronunciation, although there are two old wives tales that the 'p' meant 'prisoner', or in Ireland 'Protestant', both are incorrect. Examles of early recordings include John Thompson in the Charters of the Abbey of Whitby, Yorkshire, in 1349, and Thomas Tomson, who married Elizabeth Harris at the church of St Jon the Evangelist, Dublin, on December 12th 1631. The earliest Coat of Arms is probably the following granted in Yorkshire in 1559. Per fess silver and black, with a fesse embattled between three falcons counterchanged, belled, beaked and jessed in gold. The crest is an arm holding a gold truncheon . One of the very earliest settlers in the New World, was William Thompson recorded as 'living at Elizabeth Cittie, Virginea', before February 16th 1623. The first recorded spelling of the family name is shown to be that of John Thomson, which was dated 1318, in the "Annals of Scotland", during the reign of King Robert 1 of Scotland, known as "The Bruce", 1306 - 1329. Surnames became necessary when governments introduced personal taxation. In England this was known as Poll Tax.
Madigan
This is a famous Irish clan surname recorded as Madden, Madigan, and MacAvaddy. It derives from the pre-10th century Olde Gaelic O'Madain, translating as 'The descendant of the son of the hound'. Most Irish surnames originate from a chief's nickname, O'Kennedy, for instance means 'The descendant of the ugly headed one!' The O'Madain's originated from lands on the River Shannon in County Galway, at one time holding over 25,000 acres. Even today nameholders are still numerous in that part of Ireland. The Madigan branch of the clan are regarded as almost exclusively a Clare-Limerick family, although a branch are to be found in Counties Antrim and Derry in Ulster. Early surname holders included the Rev. Samuel Madden (1680 - 1760), a famed philanthropist, whilst Richard Madden, (1798 - 1886) was the author of the book 'The United Irishman'. During the infamous 'Potato Famine' of 1846, many name holders emigrated to both England and America. Amongst these were Walter Madden and his wife Mary, and their children Richard aged five and Alice, a baby. They sailed from Galway, bound for New York on the ship 'Junius ' on May 1st 1846. Another recording is that of Hannah Madigan aged twenty two, who left Belfast on the ship 'Howard' for New York, on July 23rd 1846. The first recorded spelling of the family name is believed to be that of Dermot O'Madadhain, which was dated circa 1100. He was chief of the Ui Maine, Connacht, during the reign of King Henry I st of England, known as 'The Just", 1100 - 1135. Throughout the centuries, surnames in every country have continued to "develop" often leading to astonishing variants of the original spelling.
Source The Internet Surname Database accessed June 5, 2010
