Gregory Lauder-Frost has carried out the most thorough research into the Lauder name. He has kindly shared this with me and you can read his views in this specially prepared article. My thanks again to him.
From Lauder in Berwickshire. Robert de Lavedre is said [erroneously] to have obtained lands in Berwickshire from Malcolm Canmore (Thomson, p.6). Sir Robert de Lauedre witnessed a charter by John de Mautelent to the Abbey of Dryburgh (Dryburgh), and William de Lawedre appears as Sheriff of Perthshire in the reign of Alexander III.
As early as 1297, the Lauders were possesors of the Bass Rock in the Firth of Forth. Robert de Lauueder, who had a charter of the lands of Colden in Barony of Dalkeith, 1316 (RHM.,I,p.17), may be Robert de Loweder, justiciar of Lothian, 1327-31 (Raine,141; RHM.,I,p.38). Another Robert de Lawdre was one of the borowis for the earl of Douglas's bounds on the Middle March, 1398 (Bain,IV,510). Robert de Lawedre and Thomas of Lawedre were merchants and burgesses of Edinburgh, 1425 (ibid.,976), and Alan of Lawadyr witnessed a charter by Stephen Fleming, master of the hospital of Soltre, 1426 (Soltre, p 52). William Lawedre was one of the conservators of truce between Scotland and England, 1451 (Bain,IV,1239), and Robert of Lawdir was one of those sent to conduct the envoys with the Princess Cecilia's dower to Edinburgh, 1477(ibid.,1445). Sandiris of Lawdyr witnessed a sasine, 1478 (Home,24), and James Lawdre had letters of denization in England, 1480 (Bain,IV,1465). William Lauder, literary forger and classical scholar, died 1771.
Ladar 1550, Laudor 1498, Laudre 1388, Lauedre 1425, Laueeder 1315, Lauther 1699, Lavedier 1333, Lawadir 1426, Laweadre 1425, Lawedire 1420, Lawedre 1413; Laweddir. The name became Lawther and Leather in Ulster.
For a different view :
Lauder, a surname said to have been originally de Lavedre. The first of the name is said to have been one of those Anglo-Norman barons who accompanied Malcolm Canmore to Scotland in 1056, and obtained from that monarch certain grants of land, particularly in Berwickshire., to which he gave his own name, being also invested in the hereditory bailieship of Lauderdale.
The surname, however, is more likely to have been derived from the Leader water, called by Camden in his Britannia, Lauder; the vale through which it flows being, from an early period, called Lauderdale. The Celtic word Laudur, signifying the lesser river, or the river which breaks forth, is thought to apply to the Leader, which occasionally, after heavy rains, overflows its banks and overspreads the neighbouring lands.
Nisbet (vol.1.p.351) says that, sometimes written Lauther, the name is local, from the town and the lands of Lauder, that is "Lower than the hills that surround it."