to grasp something with the mind: to perceive its meaning, understand it, or think it over
At its core this verb names the mind catching hold of something. In Greek from Homer onward it named the moment when looking turns into recognizing, when a person not only sees a thing but takes in what it is. The New Testament keeps that flavor: it is less "to know a fact" and more "to catch the point," to perceive with the understanding.
Its range in the New Testament runs along a single line, from bare comprehension, to deliberate reflection, to forming an idea. Sometimes it is the grasp a person fails to make, as when the disciples do not perceive the point of the loaves. Sometimes it is a call to stop and think, as in "let the reader understand." Once it is the mind's power to picture something, in "more than we ask or imagine." These are shades of one act, not separate words.
It runs close to συνίημι, the mind's putting of pieces together into a coherent whole, and the two are often paired. It differs from γινώσκω, which is knowing by experience or acquaintance, and from φρονέω, which is setting one's mind or attitude on something.
Its most striking work is in Romans and Hebrews, where it names how the mind grasps what the eye cannot see: the invisible things of God perceived through what he has made (Romans 1:20), and the created world understood by faith to have come from God's word (Hebrews 11:3). Here the word's long career in Greek thought, the mind's reach toward what lies beyond the senses, is faintly heard, though none of the writers is using it as a technical term.