The concept of redeeming was much better understood in ancient times, and the verb gā'al appears more than 80 times in the Old Testament - often referring to the redeeming of family members who had been sold into slavery to pay a debt. So people in Old Testament times totally understood what God meant when he described Himself as the One who redeems Israel. They called Him Yahweh Gā'al or Elohim El Elyon Gā'al in their songs and prayers.... or even simply Gā'al, as when Job says, "I know that my Redeemer lives." And often different words are used - like ransom or rescue.
Surprisingly, the name Redeemer hardly appears in the New Testament, but the concept is there throughout. The disappointed disciples on the road to Emmaus (Luke 24: 21) talk about how they'd really believed that Jesus was the Messiah, the One who would redeem Israel... and thankfully, later parts of the New Testament confirm that He did. The same Greek verb, lytroō, is used in other scriptures to tell us that Christ redeemed us from wickedness (Titus 2: 14) and to remind us that we weren't redeemed with corruptible things like silver and gold, but with the precious blood of Christ. (1 Pet 1: 18 - 19)
Sometimes the word ransom is used - like when Jesus Himself says (Matt 20: 28; Mark 10: 45) that the Son of Man came to give his life as a ransom for many. It seems perhaps that ransom is more readily understood than redeem by people today, but the meaning is the same: a price is paid to set someone free.
A different Greek verb, exagorazō, is used when Paul tells the Galatians that Jesus redeemed them from the curse of the Law, so that they could be adopted as God's children. There's a cost involved in adoption, and that's only a part of what it means to have God as our Rescuer and Redeemer.