Answer
If you're looking for information that shows the legitimacy of worshiping multiple times throughout the week, the early church model found in Acts may be helpful (e.g. Acts 2:46-47; Acts 6:1; perhaps also 16:5). They appear to have met daily, and to have engaged in worship as a regular part of their meetings.
If instead you are looking for information defending the idea of replacing Sunday worship with worship on another day, I think you'll have trouble establishing the case. The Bible does not explicitly declare any particular day for public worship, which may be your strongest bit of information. However, it does place the apostles in charge, and under their leadership the church seems to have established the practice that Christians observe Sunday's as holy to God (e.g. this seems to have been the meaning behind the phrase "Lord's day" in Rev. 1:11).
Current Sabbath observance on Sunday is based on such ideas as the continuing validity of the Sabbath commandment, and the freedom of God's calendar to be determined separately from any human calendar. Nevertheless, the Sabbath commandment is not a local or individual commandment -- the entire people of God were to observe the same day. If one is to argue that in particular regions the modern application of the Sabbath commandment may be observed on a day other than Sunday, then one must first overcome the idea that it is to be a corporate observance by the church worldwide. Unless you argue that the Sabbath commandment has been largely abrogated by the redemptive-historical events surrounding Christ's first advent so that it no longer has application in our age (a position which we at Third Millennium reject), I'm not sure this can be done.
Answer by Ra McLaughlin
Ra McLaughlin is Vice President of Finance and Administration at Third Millennium Ministries.