The early christians made a tradition out of meeting on "the first day of the week", which is Sunday, because Saturday is the last day of the week (you can compare this to an American calendar which start the week on a Sunday and ends it on a Saturday.)
On the first day of the week we came together to break bread. Paul
spoke to the people and, because he intended to leave the next day,
kept on talking until midnight.
and 1 Cor 16:2:
On the first day of every week, each one of you should set aside a sum
of money in keeping with your income, saving it up, so that when I
come no collections will have to be made.
A hint that John saw his revelation on a day when christians used to worship is in Rev 4 where some kind of heavenly worship service is going on (but a more solid evidence is Didaché which I refer to below). It was an ordinary day of work, but it was special because it was the day when the Lord had risen (Mark 16:2).
It's not far fetched to connect this with "the day of the Lord". In the early christian book called Didaché which is "dated by most scholars to the late first or early 2nd century" (Wikipedia) it is also obvious that "the day of the Lord" was the most important day for the christians to meet on:
But every Lord's day gather yourselves together, and break bread, and
give thanksgiving after having confessed your transgressions, that
your sacrifice may be pure. But let no one who is at odds with his
fellow come together with you, until they be reconciled, that your
sacrifice may not be profaned. For this is that which was spoken by
the Lord: "In every place and time offer to me a pure sacrifice; for I
am a great King, says the Lord, and my name is wonderful among the
Didaché, ch. 14
Put these two things together, the tradition to meet on "the first day of the week" and the tradition to meet on "the day of the Lord" and I think you have a pretty good case that they both refer to Sunday.
As a side note, it may also be observed that there is another concept in the Bible of "the day of the Lord" (Isiah 13:6, 9; Ez 13:5 etc) which should not be confused with what's going on in Rev 1:10.
As Jon points out in the comments there is another question which deals more with the transit from the Jewish sabbath (Friday night to Saturday) to Sunday as the christian worship day.