As @jasonwryan has observed, Debian GNU/Linux is engineered to do what you wish. In a comment, you have mentioned Debian's netinst image with dismay, but actually, that is just the image you seek. The image itself (though fairly small) is not minimal, but you are not going to install everything in the image onto your machine. You are going to install only a minimal part of it.
Debian organizes software in packages. Each package has a Priority, which is any of
The last two,
extra, comprise over 99 percent of the packages Debian distributes. You aren't going to install any of those, so let's talk about the other three priorities.
Packages with priority
required are, well, required. If your machine does not have them, then Debian won't work. You must install all these.
Packages with priority
important are needed that your system be usable in any normal way. You need not install these in theory, but I believe that you will install them all. They aren't many. The principal distinction between
important is that the
required packages afford your system enough functionality to be able to install (and/or uninstall) other packages including
important packages. The
required suite alone really is not enough to do anything interesting with your system (but if you want to try a
required-only system as an experiment nevertheless, feel free).
Packages with priority
standard are not needed but they are not many and are, well, standard. The usual thing to do would be to install the standard packages because even lightweight systems normally carry these, but this is up to you. I gather that you would prefer to omit the
standard. This is fine.
There are several ways to install only
important packages. To list them all here would be too much. However, this ought to get you started.
The above ignores one point: the kernel. The kernel packaged by Debian is not minimal. You can strip it down, recompile it, and make it much smaller,
but this takes some work and some trial and error. Use the
make-kpkg package for this. Be prepared to do a lot of reading before you can make this work, but it's pretty fun one you get the hang of it.
Most users aren't going to recompile their kernels, though, so, during installation, watch for a dialog re kernel drivers to appear on your screen. You may have to set your debconf priority at the start of installation to
medium, as I do, or lower; check this yourself. Look for the action "change debconf priority" on the main (not the initial) installer menu.
Once you have the debconf priority set to
medium or lower, during the "install the base system" step, a dialog will eventually appear on your screen asking if you want all drivers or only targeted ones. Choose "targeted."
MORE ABOUT THE INSTALLER
A StackExchange answer like this is going to be limited in length. It will not be a full manual. However, if unsure what to do with the installer, do the following.
- Boot your machine from the netinst image to start the installer.
- At the front-page menu, select advanced options then expert install (The level of expertise here required is moderate, but you are trying to do something nonstandard, so expert install is what you want.)
- Wait for the installer to boot.
- Before selecting any other option from the main installer menu, scroll down to set debconf priority.
- Choose priority medium.
- After choosing the priority, then, each time the installer returns you to its main menu, do not scroll up or down, but just accept the default the installer highlights in each instance. The installer will walk you through the installation steps in this way (but see also point 7, next). If unsure what I am talking about, run the installer per the above steps: you'll soon see what I mean.
- You can optionally skip over the installer stages detect network hardware and configure the network (and even configure the clock, if you don't mind using UTC time). How to skip? Answer: when the installer highlights a step you wish to skip like detect network hardware, press the DOWN key to skip it. Most of steps the installer highlights are necessary, however; there are only a few you can reasonably skip.
- Note: skipping detect network hardware does not prevent your network interface from working after installation. It just means that you won't be networked during installation.
- For questions, consult Debian's installation manual here.
There is a certain amount of this you are just going to have to figure out for yourself. Though no harder than it needs to be, Debian is not really a beginner's system. Beginners can instead learn on a Debian derivative like Mint or Ubuntu—though I do not believe that you should try that using the minimal machine you have described.