"train (an infant or animal) to forego suckling," c. 1200, from Old English wenian "to accustom, habituate," from Proto-Germanic *wanjan (source also of Old Norse venja, Dutch wennen, Old High German giwennan, German gewöhnen "to accustom"), from PIE *won-eyo-, causative form of root *wen- (1) "to desire, strive for."
The sense of "accustom a child to not suckling from the breast" in Old English generally was expressed by gewenian or awenian, which has a sense of "unaccustom" (compare German abgewöhnen, entwöhnen "to wean," literally "to unaccustom"). The modern word might be one of these with the prefix worn off, or it might be wenian in a specialized sense of "accustom to a new diet." Figurative extension to any pursuit or habit is from 1520s.
-ling
diminutive word-forming element, early 14c., from Old English -ling a nominal suffix (not originally diminutive), from Proto-Germanic *-linga-; attested in historical Germanic languages as a simple suffix, but probably representing a fusion of two suffixes: 1. that represented by English -el (1), as in thimble, handle; and 2. -ing, suffix indicating "person or thing of a specific kind or origin;" in masculine nouns also "son of" (as in farthing, atheling, Old English horing "adulterer, fornicator"), from PIE *-(i)ko- (see -ic).
Both these suffixes had occasional diminutive force, but this was only slightly evident in Old English -ling and its equivalents in Germanic languages except Norse, where it commonly was used as a diminutive suffix, especially in words designating the young of animals (such as gæslingr "gosling"). Thus it is possible that the diminutive use that developed in Middle English is from Old Norse.