Dia is a very common Greek preposition (~668 occurrences in NT). With the genitive case it means "through" — the channel or means by which something occurs. With the accusative case it means "because of," "on account of," "for the sake of."
The core image is passage or channel: water flowing through a pipe, action accomplished through an agent. In theological contexts, it frequently designates the mediatorial role of Christ and the agency of the Holy Spirit.
Dia (genitive) describes Christ as the channel of creation and redemption: "through him all things were made" (John 1:3); "through him to reconcile to himself all things" (Colossians 1:20). He is the dia-person — the one through whom God acts in every direction.
Justification "through faith" (dia pisteōs, Romans 3:22; Galatians 2:16) is the central Reformation slogan — access to righteousness is not achieved but received through the channel of faith that is directed to Christ alone.
"For the joy set before him" (dia + accusative, Hebrews 12:2) — Christ endured the cross "because of" the joy ahead. The dia-accusative gives us the motive behind the suffering: purposeful love oriented toward the restoration of joy for His people.