Here's my answer to the same question (more or less) on Christianity.SE:
Some terms might not be defined in Scripture, but this doesn't mean their meaning wasn't known or taught to the first Christians, as if the Apostles only taught by their letters, contrary to 2 Thessalonians 2:15, and to reason.
1 Corinthians 16:1 (DRB)
On the first day of the week let every one of you put apart with himself, laying up what it shall well please him; that when I come, the collections be not then to be made.
Acts 20:7 (DRB)
And on the first day of the week, when we were assembled to break bread, Paul discoursed with them, being to depart on the morrow: and he continued his speech until midnight.
Revelation 1:10 (DRB)
I was in the spirit on the Lord's day, and heard behind me a great voice, as of a trumpet,
(This appears to be a reference to a state of prayer, such as in a liturgical setting.)
From these alone we can gather that this was a set time for assembly: Sunday. The reason for the solemnity of the first day, Sunday, as a kind of new 'the Sabbath,' arose (no pun intended) in that Christ rose from the dead on the first day of the week, symbolizing a New Creation.
Mark 16:9 (DRB)
But he rising early the first day of the week, appeared first to Mary Magdalen, out of whom he had cast seven devils.
But we aren't stuck with just this. Many early Christians explicitly identify the title 'the Lord's Day' as a reference to Sunday, the first day of the week.
For example, the earliest catechism or teaching document outside of the New Testament, the Didache (~ A.D. 70-90), makes mention of the day Christians come together for the Eucharist:
But every Lord’s day do ye gather yourselves together, and break bread, and give thanksgiving after having confessed your transgressions, that your sacrifice may be pure. (XIV, 11)
(Of course, 'breaking the bread' was how the Eucharist was known in earliest days of Christianity, itself formerly known as 'the Way.')
It's significant that it's not 'on the Sabbath,' but a day the audience already know as 'the Lord's Day'—as is clear from the consistent testimony of the writings which follow, obviously Sunday, the first day of the week.
Again, Ignatius, writing still very early (A.D. 110):
If, therefore, those who were brought up in the ancient order of things have come to the possession of a new hope, no longer observing the Sabbath, but living in the observance of the Lord’s Day, on which also our life has sprung up again by Him and by His death—whom some deny, by which mystery we have obtained faith, and therefore endure, that we may be found the disciples of Jesus Christ, our only Master (Letter to the Magnesians, IX)
This of course means the Lord's Day was understood as Sunday, the day on which Christ rose from the dead.
Again, Justin Martyr, an important early Christian apologist (writing here appealing to the Romans under whom they were persecuted about the benignity and innocence of the Christian faith):
And we afterwards continually remind each other of these things. And the wealthy among us help the needy; and we always keep together; and for all things wherewith we are supplied, we bless the Maker of all through His Son Jesus Christ, and through the Holy Ghost. And on the day called Sunday, all who live in cities or in the country gather together to one place, and the memoirs of the apostles or the writings of the prophets are read, as long as time permits; then, when the reader has ceased, the president verbally instructs, and exhorts to the imitation of these good things. Then we all rise together and pray, and, as we before said, when our prayer is ended, bread and wine and water are brought, and he that presides in like manner offers prayers and thanksgivings, according to his ability, and the people assent, saying Amen; and there is a distribution to each, and a participation of that over which thanks have been given, and to those who are absent a portion is sent by the deacons. And they who are well to do, and willing, give what each thinks fit; and what is collected is deposited with the president, who succours the orphans and widows and those who, through sickness or any other cause, are in want, and those who are in bonds and the strangers sojourning among us, and in a word takes care of all who are in need. But Sunday is the day on which we all hold our common assembly, because it is the first day on which God, having wrought a change in the darkness and matter, made the world; and Jesus Christ our Saviour on the same day rose from the dead. For He was crucified on the day before that of Saturn [Saturday]; and on the day after that of Saturn, which is the day of the Sun, having appeared to His apostles and disciples, He taught them these things, which we have submitted to you also for your consideration. (First Apology, LXVII)
(Justin defines elsewhere that 'the Gospels' are what he means by 'memoirs of the apostles,' this being a condescension to non-Christians who know nothing of the Faith.)
'The Day of the Lord,' or 'the Day,' is a reference to the summation of all things, the end of days, the Judgement proper of all.
This is the sense in which this day is the day of the Lord. It is quite distinct from that future Day spoken of in, e.g., 1 Corinthians 3:13.