Oh my lordy-- there are so many different "open-source" wiki programs on the market. It is overwhelming. There is even a site dedicated to allowing you to compare them -- or at least the 30 or so that they have. Go to the bottom left corner and there is a popup box where you can choose for those you want to compare with:
To their list I add:
PHP Wiki's Easy Installation
PhP Wiki is free and open source. Description:
A WikiWikiWeb is a web site where
anyone can edit the pages through an
HTML form. Linking is done
automatically on the server side; all
pages are stored in a database.
It can also be one click installed through Fantastico which a script installing program supplied through cpanel which is the most common software managing an Apache server account. Most places that you purchase share hosting or a managed hosting on an Apache/Linex server provide Cpanel, and thus support a one click installation of PhP Wiki.
It is simple, clean, no frills and does just what I am hearing that you would like.
The Wikipedia Engine
If you are attacted to Wiki's because of wikipedia -- why not use the same engine? MediaWiki is easily downloaded and installed onto an internal server. Or your server admin can install it on a remote server. There is far more support including videos and user guides for how to get going on their site. Very easy to use. And free. It API has a developer API if/wehn you grow and want to integrate it to other functions of the company.
Google Docs as an option?
While technically not a wiki -- and all of the good wiki purest will be outraged at the thought of it being considered at the same time: Google Doc is another option to consider. A shared file system -- like Google Docs that tracks discussions and version control might also meet your business need. You can share docuement, set read/write permissions and now their new "Discussions" feature allows team members to discuss their ongoing development of the docuement. It also has a good built in version control allowing you to restore back to any point of the process. The benefit is the connection with your other services, low cost (free) and relativly low learning curve.
Wiki + DropBox
And en additional interesting solution is deploying TiddleyWiki onto a shared drop box folder. No need for a server at all. It provides a basic wiki environment for your collaborative development of shared files -- without the need for a server -- or even a connection to the server to access. Here's a guide on how to do it.
If your website is built in one of the common open source CMS like Joomla, Drupal or Wordpress there is an integrated wiki extention which will allow for integration with your website structure. This allows you site to have an "intranet" component. Encouraging social media development, regular visiting your own website, accessiblity might be added value reasons you would want to host your wiki integrated with your site. Here is the example of the Joomla wiki list.
Good Luck. I hope that is not too overwhelming. I had just done this research for a client and I am glad someone else will be able to take advantage of it!