I can answer this! A GNU/Linux distribution is an amalgamation of software pieces from GNU and elsewhere. At the very least, you want to fork the software packages you wish to modify, modify them, recompile them, and then redistribute your fork of the distribution of choice.
That wouldn't be all however, as you have to be very careful with the trademarks and copyrights of the distribution you are forking. (I am not a lawyer) but it would be advised to change the branding of the distro you are forking, as you (probably) don't want to associate your distribution with the parent one (I'm not sure about that one, but it might create issues with the parent distribution. If you want to say that you forked a specific distribution explicitly, you can mention it explicitly, but don't use their trademarks. Again, I am not a lawyer).
You also have to find a way to provide a repository with software for your distribution, and also figure out if you want your repositories to exist besides the main distribution's ones or be the only ones for your distribution, and also find out ways to not break the distribution when someone updates it.
Apart from that, there are also 3 files you may explicitly want to change, them being
/etc/lsb-release, with the last one need to be changed to change the GRUB listing for your distro. Be sure to checkout
/etc/default/grub for the line
GRUB_DISTRIBUTOR. It should be something like
GRUB_DISTRIBUTOR=lsb_release -i -s 2> /dev/null
Now, how you would go about software wise is entirely up to you (subtracting and adding software), but your new distribution should be ready by now (I don't know if I forget anything at this point).
Still however, if you want my personal opinion, I would rather create a new linux distribution following Linux from Scratch, and then throw a package manager on top, rather than directly fork a distribution. Forking a distribution has the advantage that you get a headstart, but if you decide to remove things, it might become a headache very very fast, as you can never be sure about dependencies in the parent distribution.