Hi,
Most distro's are rather over-complete, but that you already noticed, otherwise you wouldn't have asked this specific question
You could use LFS and a few needed packages from BLFS and/or some none-LFS packages. You will end up with a machines that has the basics to run linux (LFS) and the dependencies needed for your own application. That's about as lean as it can get.
Pro's: Very lean. Besides having installed only those packages that are needed, everything is compiled for your specific hardware.
Con's: You need experience to set up/configure LFS.
If building your own is not your thing, you can always check
distrowatch for a distro that fits your needs.
If security is an issue and you don't have the experience to set that up from scratch, you can take a look at
DW: Slackware and
DW: (free)BSD
Both are solid, stable and secure. They are not 'state of the art', they rely on packages that have proven themself (for example: Slackware uses kernel 2.4, not 2.6).
You did ask for a linux solution, that said: freeBSD is not linux, it's unix. Don't know if your application allows that.
Hope this clears things up a bit.