early 14c., allouen, "to commend, praise; approve of, be pleased with; appreciate the value of;" also, "take into account or give credit for," also, in law and philosophy, "recognize, admit as valid" (a privilege, an excuse, a statement, etc.). From late 14c. as "sanction or permit; condone;" in business use from early 15c.
The Middle English word is from Anglo-French alouer, Old French aloer, alloiier (13c.) "place, situate, arrange; allot, apportion, bestow, assign," from Latin allocare "allocate" (see allocate). This word in Old French was confused and ultimately merged with aloer; alloer "to praise, commend, approve," from Latin allaudare, adlaudare, compound of ad "to" (see ad-) + laudare "to praise" (see laud).
Between the two primary significations there naturally arose a variety of uses blending them in the general idea of assign with approval, grant, concede a thing claimed or urged, admit a thing offered, permit, etc., etc. [OED].
From the first word came the sense preserved in allowance "money granted;" from the second came allowance "permission based on approval." Meaning "assert, say," 19c. U.S. colloquial, also was in English dialect and goes back to 1570s. Related: Allowed; allowing.
late 14c., "to refuse to praise" (a sense now obsolete), from Old French desalouer "to blame," from des- "not, opposite of" (see dis-) + alouer (see allow). Meanings "to reject, refuse to receive or acknowledge," also "refuse to allow, refuse to approve or sanction" are from c. 1400. Related: Disallowed; disallowing; disallowance.
Meaning "sanction, approval, tolerance" is from 1550s. Sense of "a sum allotted to meet expenses" is from c. 1400. In accounts, meaning "a sum placed to one's credit" is attested from 1520s. Mechanical meaning "permissible deviation from a standard" is from 1903. To make allowances is literally to add or deduct a sum from someone's account for some special circumstance; figurative use of the phrase is attested from 1670s.
early 15c., permitten, transitive, "allow (something) to be done, suffer or allow to be," from Old French permetre and directly from Latin permittere "let pass, let go, let loose; give up, hand over; let, allow, grant, permit," from per "through" (from PIE root *per- (1) "forward," hence "through") + mittere "let go, send" (see mission). Meaning "grant liberty or leave" is from 16c. Related: Permitted; permitting.