I think that one of the most persuasive points to make in being favor of democracy is the idea that it is the form of government in which the most amount of voices can be heard. If one starts with the premise that silencing voices is the worst thing a government can do, then democracy is the form of government in which most voices can be heard. While the contention is that democracy does not get much done in the way of ensuring action, it does ensure that as many voices as possible are heard in the process of deliberation and understanding on a topic. This would be one point in favor of democracy.
Another point in favor of democracy would be to point to the flaws in other forms of government. Going back to the idea that silencing voice is the worst element in any government, other forms of government such as fascism or communism do a good job of silencing some voices at the cost of raising others. In both theory and practice, liberal democracy is designed to hear voices, integrating them into the fabric of governance. This would make it fundamentally more desirable than other forms of government present. Its flaws aside, its ability to hear voices and include them into the process of governance would be a strong point to make in being in favor of democracy.
I would like to add a few points to the previous answer. In addition to giving the most people a voice in government a democracy promotes involvement. If the people know that their voice can and will be heard, then those people are actively involved with speaking up or even becoming a government leader. On the other side of the vote a democracy can create a sense of obligation in the elected officials. Elected officials will likely have a sense of obligation to the people that voted them into office. The elected officials will want to keep their constituents happy, so that elected official will work toward creating policies and goals that benefit the voting citizens.
A democracy also helps to establish equality. Since everybody can vote and everybody's vote is worth the same amount, a democracy works to form equality among the people of that government.
I think one huge advantage of a democracy is that it does not place all of the power in the hands of a single individual. A group distributes the power, which works to limit exploitation of citizens. There is also less of a chance of corruption.