What does dayspring mean in the KJV Bible?

The word "dayspring" is an archaic term meaning "dawn" or "morning." It is used in Job 38:12 and Luke 1:78 in the KJV.

Job 38:12 (KJV) says, "Hast thou commanded the morning since thy days; and caused the dayspring [Heb., shachar] to know his place." The verse concerns God's sovereign all encompassing power and majesty. God even uses the dawn according to his will. God is immanent; he interacts with his creation everyday. He is also transcendent, as he literally controls the dawn (including its aurora, beauty, et. al.) in its constant varying its position. See Genesis 1:3; Psalms 65:8; Amos 5:8.

Luke 1:78 (KJV) says, "Through the tender mercy of our God; whereby the dayspring [Gk. anatole] from on high hath visited us," It refers to the Messiah, who is the light of men (cf. Isa 60:1; Mal 4:2; John 1:4-5, 9-10; Eph 5:14; 2 Pet 1:19). Some translations use the word "Sunrise" (Berean Literal Bible, NASB, NAS) or "Dawn" (Berean Study Bible, Holman Christian Study Bible). The dawn is seen in the east rising upward and Jesus will return from the east (Matt 24:27). As the darkness speaks to us of our depravity, so Christ, as the Dayspring illumines us with "the light of the knowledge of the glory of God in the face of Jesus Christ" (2 Cor 4:6). The Septuagint (Greek version of OT) uses anatole in Jeremiah 23:5 translated as "Branch" (cf. Isa 4:2 - Christ is a sprouting branch and a rising star).