word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).
The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).
small insectivorous mammal, Old English screawa "shrew-mouse," unknown outside English, and "the absence of evidence for the word between the OE. period and the 16th c is remarkable" [OED]. Perhaps from Proto-Germanic *skraw-, from PIE *skreu- "to cut; cutting tool" (see shred (n.)), in reference to the shrew's pointed snout. Alternative Old English word for it was scirfemus, from sceorfan "to gnaw."
The meaning "peevish, malignant, clamorous, spiteful, vexatious, turbulent woman" [Johnson] is late 14c., from earlier sense of "spiteful person" (male or female), mid-13c., traditionally said to derive from some supposed malignant influence of the animal, which was once believed to have a venomous bite and was held in superstitious dread (compare beshrew). Paired with sheep from 1560s as the contrasting types of wives.