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Waltheof had one illegitimate son by an unknown mistress:
From http://fmg.ac/Projects/MedLands/SCOTTISH%20NOBILITY.htm#GospatrickD...
d) GOSPATRICK . The Cronicon Cumbriæ records that “Alanus filius et hæres eiusdem Waldevi” had “unum fratrem bastardum…Cospatricium” to whom he granted property[1450]. A memorandum dated to [1275] records that "Alan", son of "Earl Cospatryk formerly earl of Dunbar", had "a bastard brother on his father’s side…Cospatryk to whom he gave the township of Bolton within the forest, and Bastenthwyt outside the forest, and Hestholm in Derwentwatre"[1451]. "Alexander nepos regis Alexandri, Beth comes, Gospatricius Dolfini, Mallus comes, Madach comes, Rothri comes, Gartnach comes, Dufagan comes, Willelmus frater regine, Edwardus constabularius, Gospatricius filius Walthef, Ufieth Alfricus pincerna" witnessed the charter dated to [1114/15] under which "Alexander…rex Scottorum filius regis Malcolmi et regine Margerete et…Sibilla regina Scottorum filia Henrici regis Anglie" reformed Scone Abbey[1452]. A charter of King Henry II records donations to York St Mary, including the donation of “villam de Stainburn” by “Walthef filius Gospatrici” and "villam de Saltergh et…de Whyrkingtona" by "Gospatricius filius eorundem"[1453]. "…Gospatricius filius Waltheof…" witnessed the possibly spurious charter dated to [1120] of "Alexander…Rex Scottorum…Sibilla regina Scottorum…"[1454]. "…Gospatricio filio Waltheui" witnessed the charter dated 1124 under which "Alexander…Rex Scottorum" granted jurisdiction to the prior of Scone[1455]. "…Alano filio Waldeof et Gospatrico fratre suo…" witnessed a charter dated 1139 under which "David Rex Scotie" confirmed the grant of Coldingham by "Gospatricus comes frater Dolfini" to St Cuthbert[1456]. “Waldevus filius Cospatrici comitis” donated property to Gysburn Priory, with the consent of “uxore mea Sigrida et filiis meis Cospatrico et Alano”, by undated charter[1457]. 1156. m ---. The name of Gospatrick’s wife is not known. Gospatrick & his wife had [one child]: ..