The name Hagar in the Bible
Hagar is the mother of Ishmael, the first son of Abram, later Abraham (Genesis 16:1). When his wife Sarai, later Sarah, realizes that her own scheme successfully results in Hagar's pregnancy, she pesters her until she leaves. Hagar meets the Angel of YHWH at the well of Beer-lahai-roi, and He sends her back. When Sarah gives birth to Isaac, the ancestor of Israel, she objects to Hagar and her son Ishmael, and a heavy hearted Abraham sends her away for good.
Hagar travels south, possibly to go home to Egypt, but loses her way. The verb used to describe her "wandering" about in the wilderness of Beersheba is תעה (ta'a), to err, go astray. A derivation of this verb is תועה (to'a), error.
Hagar wanders the desert until her supplies run out. Desponded and exhausted, Hagar abandons her son so that she won't have to watch him die, and sits down a bow shot away from the boy to cry. Curiously, not her cries but the cries of the boy reach heaven, and God shows up. For the second time the Angel of YHWH speaks to Hagar, and promises her that Ishmael will be a great nation. God opens her eyes and she sees the well of Beersheba. This is fortunate for two reasons. First of all she and Ishmael now have water, but they also know again where they are.
Quickened Hagar and Ishmael resume their journey. Ishmael becomes an archer and lives in the wilderness of Paran (in the Sinai desert between the Gulf of Suez and the Gulf of Aqaba) with his Egyptian wife whom Hagar has obtained for him.
Hagar becomes the grandmother of twelve princes, who form the great nation that God promised. That nation is not really named in Scriptures but their function may be explained by the amazing parallel of The Standard Model of Elementary Particles and the family of Abraham. Ishmael's sons do not become the rivals of Israel as one may expect. His son Kedar gets mentioned for their signature black tents, once positive in the Song of Solomon (1:5), and once negative by king David in Psalm 120:5. The prophet Isaiah mentions Ishmael's first born son Nebaioth and his brother Kedar among the nations that will be gathered up into the Kingdom of God (60:7).
The apostle Paul compares Hagar (Αγαρ; Agar) to one of two covenants, namely the covenant of slavery that Jerusalem has with the Law that went forth from mount Sinai. The heavenly Jerusalem, on the other hand, is free and her children are Isaac; the children of the free woman, and of the promise (Galatians 4:24).
Etymology of the name Hagar
Although Hagar is an Egyptian, her name is Semitic. It comes from the verb הגר (hgr), which is not used in the Bible and its meaning is subsequently unknown. NOBSE Study Bible Name List and HAW Theological Wordbook of the Old Testament insist that the verb means to flee. BDB Theological Dictionary refers to the Arabic equivalent, meaning forsake or retire. An Arabic noun derived of this root serves as the name for Mohammed's famous flight, the Hegira. Another name derived of this same root is the tribal name of the Hagrites (Psalm 83:7, 1 Chronicles 27:31).
A curious detail is that the reticent root הגר (hgr) looks like a construct of the definite article ה (ha), and a trace of the verb גרר (garar), drag out or away:
The name Hagar means Flight thanks to a very rare verb, but is also highly similar to the more common words The Sojourner/ The Dragged Away One.