to make firm and fixed: to set something solidly in place, and so to steady a person so that they hold firm
At its core this word means to make something firm and immovable. Its most concrete picture in the New Testament is the great gap between the rich man and Lazarus, a chasm "fixed" so solidly that no one can cross it. From that physical image the word reaches in two directions.
The first is human resolve: Luke writes that Jesus "set his face" toward Jerusalem, fixing his purpose so firmly that nothing would turn him back. The second, and by far the most common, aims that same firmness inward. God, or a fellow believer, makes a person steady so that their faith, their heart, and their resolve will hold when they are under pressure. This is the word Jesus gives Peter when he tells him to "strengthen" his brothers, and the word Paul and Peter reach for again and again when they pray that a young church will not be shaken.
Because of this, the word sits close to several others without being any of them. βεβαιόω confirms that a promise or testimony is reliable; θεμελιόω grounds something on a foundation, the work of first building it; στερεόω makes a body or thing physically solid. This word is the bracing that keeps what already stands from toppling over.