Old English dal "state of being divided; a sharing, a giving out" (original senses now mostly obsolete), shortened from gedal "portion," and related to dæl "deal," from Proto-Germanic *dailan (source also of Old Frisian and Old Saxon del, Middle Dutch deil, Dutch deel, Old High German teil, German Teil "part, portion"), which is said in Watkins to be from PIE *dail- "to divide," a Germanic-Slavic extended form of the root *da- "to divide." But Boutkan writes, "Most probably, we are dealing with a substratum word in Gmc."
Meaning "a part apportioned or divided out" is from late 12c.; that of "distribution of alms or gifts" is by late 13c. Specifically as "a portion of money, food, or other things distributed in charity" is by late 15c. From 1919 it became the popular name in Britain for various government payments made regularly to the unemployed. Hence on the dole (1920s).
mid-15c., dolen, "distribute, hand out (alms)," from dole (n.). Meaning "give in portions or small quantities" (as alms to the poor) is from 1749. Related: Doled; doling.