There are at least three theological foundations for common grace.
First, for it to be grace, it must be founded on the atoning work of Christ. McCune comments:
There is no one verse that anchors common grace in the atonement of Christ. However, theologically this is necessarily so. Any mitigation of the effects of sin is due ultimately to the cross work of Christ. There is no other basis on which God could deal with sin in grace or mercy. Common grace is grace—non-redemptive grace—and is a mitigation of the full effects of sin.
Rolland McCune, A Systematic Theology of Biblical Christianity
A confirmation of this is found in the fact that the Noahic Covenant, a common grace covenant, is based on a burnt offering, which Leviticus 1:4 identifies as a sacrifice that makes atonement.
Second, common grace is founded on the reality of God’s sovereignty over all creation. The good things that happen to people, the growth of food, etc. are not merely natural occurances. They are gifts from God. As Kuyper notes,
If God is sovereign, then his Lordship must remain over all life and cannot be closed up within church walls or Christian circles. The extra-Christian world has not been given over to satan or to fallen humanity or to chance.
Abraham Kuyper, “Common Grace,” in Abraham Kuyper: A Centennial Reader
Third, common grace is rooted in the reality that God built his law into creation. Living in the world contrary to God’s law always has consequences. As humans reckon with those consequences, sin is retrained to a degree. Al Wolters comments on this reality:
The structure of all the creational givens persists despite their directional perversion. That structure, anchored in God’s faithfulness, sets a limit on the corruption and bondage wrought by evil. . . . Ignoring the law of creation is impossible. The law is like a spring that can be pressed down or pushed out of sight only with great effort and that continues to make its presence felt even when repressed for a long time. The ‘structure’ of a thing is the law that is in force for it, and not amount of repression or perversion will ever succeed in nullifying its presence and effect.
Albert M. Wolters, Creation Regained