I think congregationalism is the Scripturally mandated form of church order. To clarify, by congregationalism I mean that way of ordering a church whereby ultimate decision making is vested in the local congregation, guided and led by its elders and served by its deacons. There are two types of argument I would advance in favour of this position, and since they are the two types of argument which I think are important in any theological discussion, I wanted to outline how they work. I'm only hinting here at the structure of the particular arguments themselves; mainly I'm trying to show how theological argument needs to work.
The first sort of argument is called 'the Bible says...' This sort of argument is not difficult to understand. In the case of congregationalism, it would consist of pointing out a few key passages which describe church order in the first congregations. We could point out that Scripture describes and prescribes the appointment of elders and deacons, and mentions no other church officers. We could also point out that in several instances Scripture points towards the whole congregation being involved in decision making - for example, church discipline in Matthew 18:17. I would also want to go to Acts 20, and see how Paul, foreseeing his absence, commits the churches to the word of God and not to any other officer or group of officers standing above the local congregation. In short, the Bible says that churches are run congregationally.
For many Christians, I guess that's the end of the conversation. But I do not think that any solid case can be built this way. It is one thing to be able to quote the text of Scripture, and quite another to be able to show how it applies today. There is a need to show why the Bible says what it does, and for that we have to go behind the apostolic teaching to see how it relates to the central concerns of the gospel. I am assuming here that the Biblical authors are theologians, and I am assuming a particular understanding of how they thought and wrote. I do not think we are to imagine all of the Bible being direct oracles, written down. I don't think Paul got his teaching about eldership directly from heaven. I think it is a reflection on the gospel - on the birth, life, death, resurrection, and ascension of Jesus. So what the Bible says hangs together. It is all related to the gospel, and understanding how it is related will help us to understand how it applies now. The connection is not always spelled out, or sometimes even hinted at. But we can try to think from the starting point - Christ - to the end point - this specific teaching - and work out what goes in between.
As an aside, this is why it is very hard to work out how to apply passages of the NT which seem to have no connection to the gospel, or a connection which is now obscure. I am thinking of women covering their heads, for example.
When it comes to congregationalism, I think there is a Christological and a Pneumatological point to make. The Christological point is that Christ runs his church. Jesus the King governs his church. This has implications for how we understand church order. For starters, we can't make it up - it is not up to us to derive structures which Christ has not mandated. Moreover, we must have structures that reflect the fact that Christ is actively involved in the church, and leave us open to his guiding. I think congregationalism makes sense in this context. The Pneumatological point is that every Christian has the Spirit. The officers of the church do not have a monopoly on wisdom, or on hearing from God. Congregationalism seeks to reflect this.
It is the combination of the two arguments that wins me over. The Bible says it, and I can see that what the Bible says makes sense in the light of the gospel. The application in this case is straightforward; since nothing important has changed in the circumstances surrounding the question, the Biblical guidance and instruction stands as it is.