"Autonomy" is a compound word, composed of autos, meaning "self," and nomos, meaning "law." An ordinary dictionary will tell us the word means "self-ruled," so that an autonomous church is "self-governed, without outside control."
There are those who reject the concept of God and revelation, saying ultimate authority is in man. To them there would be no limitations placed upon self-rule. Of course most of our readers accept Christ as King, and know that a church which wishes to exercise "self rule" in all things is not the church of Christ. But our brethren are far from clear on the legitimate (scriptural) field of self rule, and how this affects the relation of one church to another. Some seem to think "autonomy" means the right to devise organizational arrangements for which there is no New Testament authority; while others think calling attention to such error violates the "autonomy" of the erring brethren.
A church can not "rule" on the importance of Christ' death, the necessity of faith, the meaning and purpose of baptism; for these are legacies of truth which Christ gave the world and by which we are called. The church is the product of the gospel, not its author. One would not violate some church's autonomy by teaching along these lines, for no church as a legitimate "say" in such things.
Does God give a local church the right to decide the day of worship? May they "rule" on the need for assembling, or the so-called "items" of acceptable worship? It is not clear that even in those things assigned as church (team) activity, a distinction must be made in that which is part of "the faith, once for all delivered unto the saints" (over which the congregation has no rule), and such details as are left to human judgement. The field of church autonomy is that of human judgement, and that only.
As an example: God's word indicates the day on which saints are to partake of the Lord's Supper -- but it does not specify the time of day. The time is left to human judgement, and therefore to the "rule" of brethren. A church exercises autonomy when it sets its own time of assembling -- and we might add, that time rests upon human authority, not upon divine mandate. Each church has this same right and may choose different times. If one sought to unduly influence or alter another's time of meeting, this would be interfering with "autonomy."
But if one church should declare Thursday the Lord's Day, others could seek to teach them more perfectly the way of the Lord -- and violate no legitimate "autonomy" in doing so -- for no church has the scriptural right to "rule" in matters God has settled.
When brethren have honest differences in their understanding of what God has said, one church may believe their "ruling" is done in matters of judgement, while another may believe they violate plain teachings of God. If both parties are equally interested in serving God, neither will rest the case in "our rights," but will be happy to study God's word together so that God can rule supremely in all.