Web-based (intranet / non-hosted) timesheet / project tracking tools
Achievo: price certainly can't be beat - and it fills the other requirements (so far) of my initial list.
I've now had a chance to use (or try to use) both Achievo and Collabtiv, and must say that Collabtiv is non-ideal. I couldn't get it setup at all, even after going through all the docs on the website and that come with it.
I'm running RHEL 5 x64 with PHP 5.2 and MySQL 5.0, and Achievo setup with no problems. For now, this looks like the solution I'll go with.
Collabtiv: this one also can't be beat form the price perspective, and also matches (so far) my requirements list.
However, I cannot get it up and running on my RHEL 5 x64 box (PHP 5.2 and MySQL 5.0). Don't know what's up with it, as the previews online looked really really promising.
For now, I'm going with Achievo.
The Ruby on Rails application ClockingIT sounds like you could need it.
It's not quite using the LAMP stack, but the installation should be quite easy on any reasonably modern Linux distribution which provides packages for Ruby (and ideally Ruby on Rails).