"Give Me Your Hand" is a tune from the early 17th century by Ruairí 'Dall' Ó Catháin, (blind Rory O' Cahan, c.1570-c.1650), perhaps in honour of a lady. It is one of the most widely recorded pieces of Irish and Scottish traditional music.
"Give Me Your Hand" is also the title of many another song

Song

In music, a song is a composition for voice or voices, performed by singing.A song may be accompanied by musical instruments, or it may be unaccompanied, as in the case of a cappella songs...

s; e.g. Dorothy Stewart

Dorothy Stewart

Dorothy Newkirk Stewart was an American printer, printmaker and artist. She was boron April 8, 1891 in Philadelphia to Dr. William Shaw and Delia Allman Stewart. Her parents sent her and her sister Margretta to private school in Philadelphia.-Art Training:...

's unrelated song is well known.

Ruairí Dall O'Catháin

According to Edward Bunting

Edward Bunting

Edward Bunting was an Irish musician and folk music collector.-Life:Bunting was born in County Armagh, Ireland. At the age of seven he was sent to study music at Drogheda and at eleven he was apprenticed to William Ware, organist at St. Anne's church in Belfast and lived with the family of Henry...

, in The Ancient Music of Ireland, this harp tune was written in about 1603 by Ruairí 'Dall' Ó Catháin (blind Rory O' Cahan).

This tune, revived by Sean O'Riada, was originally a composition of the blind Derry harpist Rory 'Dall' O'Cathain. He wrote it while (in) Scotland, where he had a disagreement with a Lady Eglington. He composed the tune for her when she apologized.

from the Wolfetones.

Captain Francis O'Neill

Francis O'Neill

Francis O'Neill was an Irish-born American police officer and collector of Irish traditional music.O'Neill was born in Tralibane, near Bantry, County Cork. At an early age he heard the music of local musicians, among them Peter Hagarty, Cormac Murphy and Timothy Dowling. At the age of 16, he...

 suggests

Proud and spirited, he resented anything in the nature of trespass on his dignity. Among his visits to the houses of Scottish nobility, he is said to have called at Eglinton Castle, Ayrshire. Knowing he was a harper, but being unaware of his rank, Lady Eglinton commanded him to play a tune. Taking oftence at her peremptory manner, O'Cahan refused and left the castle. When she found out who her guest was her ladyship sought and effected a speedy reconciliation. This incident furnished a theme for one of the harper’s best compositions. “Tabhair Dam Do Lamh,” or “Give Me Your Hand!” The name has been latinized into “Da Mihi Manum.”
The fame of the composition and the occasion which gave birth to it reaching the ear of King James the Sixth, induced him to send for the composer. O’Cahan accordingly attended at the Scottish court, and created a sensation.

There are other stories and legends, but no authoritative evidence or references are known.

The tune is sometimes claimed to be written by the famous harpist [O'Carolan], who lived some years later. However there is no reference in the Bunting collection of O Carolan’s music. Nor should Ruairí 'Dall' Ó Catháin be confused with another blind poet at around the same time, Rory 'Dall' Morrison.

A number of apocryphal stories have circulated about the circumstances of the tune's composition; further details can be seen at Andrew Kuntz's The Fiddler's Companion.

Later references to the tune

The Fiddler's Companion says

The Latin title first appears in the Wemyss manuscript of 1644 and in the Balcarres manuscript of 1692

and then

The melody's popularity was long-lived, as attested by its appearance in many collections throughout the 18th century, including Wright's Aria di Camera (1730), Neal's Celebrated Irish Tunes (c. 1742—a revised date from the oft-given 1721 or 1726, this based on watermark research—see the appendix to the 2001 edition of O’Sullivan’s Carolan), Burk Thumoth's Twelve English and Irish Airs (c. 1745-50), Thompson's Hibernian Muse (c. 1786), Brysson's Curious Selection of Favourite Tunes (c. 1790, and Mulholland's Ancient Irish Airs (1810).

English and Gaelic titles first seem to have appeared in 'A Collection of Ancient Irish Airs', by John Mulholland, 2 vols. Belfast, 1810.

Seán Ó Riada

Seán Ó Riada

Seán Ó Riada , was a composer and perhaps the single most influential figure in the revival of Irish traditional music during the 1960s...

 is attributed with reviving the tune in the late 1960s. The Wolfetones also contributed to the tune's development by adding words of reconciliation at a time of violence in Northern Ireland

Northern Ireland

Northern Ireland is one of the four countries of the United Kingdom. Situated in the north-east of the island of Ireland, it shares a border with the Republic of Ireland to the south and west...

 during the The Troubles

The Troubles

The Troubles was a period of ethno-political conflict in Northern Ireland which spilled over at various times into England, the Republic of Ireland, and mainland Europe. The duration of the Troubles is conventionally dated from the late 1960s and considered by many to have ended with the Belfast...

.

Some recordings of "Give Me Your Hand" (Tabhair dom do Lámh)

The Wolfe Tones are the only known recordings with vocals.

• Áine Minogue & Druidstone - Tabhair dom do Lámh [The Vow-04](1998)
• Bandari - Tabhair dom do Lámh (Give me your Hand) [Garden Of Dreams-08] (1999)
• Celtic Orchestra - Tabhair dom do Lámh [Classic Celtic Moods, CD3 - 02]
• Celtic Southern Cross - Give me your Hand [Begged Borrowed & Stolen, CD3 - 26]
• Ceoltóirí Chualann (with Seán Ó Riada)- Tabhair dom do Lámh [Ceol na nUasal - 08] ( 1967)
• Ceoltóirí Chualann - Tabhair dom do Lámh [Ó Riada - 11] (1971)
• Charles Guard - Tabhair dom do Lámh [Avenging & Bright - 08] (1991)
• Cobblers' Last - Peggy & the soldier, Give me your hand [Boot in the Door - 04]
• Comhaltas Ceoltóirí Éireann - Tabhair dom do Lámh [Foinn Seisiun Book 1, p. 42 - 19]
• Dancing Willow - Tabhair dom do Lámh, Give me your Hand (see recording link above)
• Emily Cullen - Tabhair dom do Lámh [Maidens Of The Celtic Harp - 14] (1997)
• George Winston - Tabhair dom do Lámh (La Valse...Jeunes Filles) [Plains-04] (1999)
• Innisfree Ceoil - Tabhair dom do Lámh [Celtic Airs, CD1 - 02] (1996)
• James Galway & The Chieftains - Give me your Hand [05]
• Kate MacLeod & Kat Eggleston-Tabhair dom do Lámh [Drawn From The Well-09] (2002)
• Kim Robertson - Give me your Hand [Wind Shadows, Vol. I - 10] (1983)
• Lifescapes - Give me your Hand [Celtic Mystery - 08] (2000)
• Oliver Schroer - Tabhair dom do Lámh (Give me your Hand) [Celtic Devotion-09] (1999)
• Patrick Ball - Give me your Hand [Celtic Harp - The Music of Turlough O'Carolan - 10] (1983)
• Planxty - Raggle Taggle Gypsies, Tabhair dom do Lámh [Reunion Point Theatre, CD2-08](2004)
• Planxty - Raggle Taggle Gypsy, Tabhair dom do Lámh [Planxty - 01] (1973)
• Planxty - Tabhair dom do Lámh [Vicar Street, Dublin - February 2004]
• Pól O'Ceallaigh - Tabhair dom do Lámh [Celtic Drones - 11] (1993)
• Rosemary Beland - Tabhair dom do Lámh [The Tinkers' Wedding - 12] (1992)
• Spailpin (with Colum Mac Oireachtaigh) -White, Orange & Green, followed by Tabhair dom do Lámh [Whiskey in the Jar-08] (1991)
• The Chieftains - Tabhair dom do Lámh [The Chieftains 5 - 02](1975)
• The Chieftains - The Cloak, Tabhair dom do Lámh [The Essential Chieftains (CD1) - 17] (2006)
• The Chieftains & James Galway - Give me your hand [In Ireland - 05] (1987)
• The Irish and the Scotch - Give me your Hand [Open Folk - 04] (1999)
• The Rambling Irishmen - White, Orange & Green & Tabhair dom do Lámh [Songs of Old Ireland-01]
• Wolfe Tones - Tabhair dom do Lámh, Give me your Hand [Till Ireland a Nation - 13] (1974)
• Wolfe Tones - Tabhair dom do Lámh, Give me your Hand[25Th Anniversary,CD1-12](1991)