"Now on the first day of the week, when we were gathered together to break bread, Paul began talking to them, intending to leave the next day, and he prolonged his message until midnight" (Acts 20:7).
"I [John] was in the Spirit on the Lord's day, and I heard behind me a loud voice like the sound of a trumpet" (Revelation 1:10).The passage in Acts has the Christians gathering together on the first day of the week to break bread. The grammar here makes this out to be a regular occurrence. And so, this, if not the day during which the Church was observing the Sabbath, needs to be explained. Also, John, being in the Spirit on the Lord's day demonstrates that the first day of the week bore enough significance to merit such a familiar nomenclature as "the Lord's day" and John speaks of this day as if it were a normal occurance. These are just some reasons.
Here are a couple more based upon Biblical Theological models for interpretation. The original Sabbath is based in the seven days of the Old Creation: God worked six days and then rested on the LAST day of the week. Whereas the Sabbath falling on the last day of the week was indicative of the Old Creation, the Christian practice of observing the Sabbath on the FIRST day of the week is a congruent with God's New Creation. Christ rose on the first day of the week and began His Sabbath rest then (cf. Hebrews 4).
Also of interest is the fact that the two versions of the Sabbath
are typological of the two covenants that go along with each
Creation. With the Old Creation, Adam was given a covenant of
works whereby he would work for a time and then receive his
heavenly rest. Adam failed in this and God uses the institution of
the Sabbath falling on the first day of the week to demonstrate that
with His New Creation, man begins in his rest and the good works follow.
These are all fair reasons I think that we, as Christians, celebrate the Sabbath on the Lord's day (i.e., Sunday).