Which OS can we install in Asus EeeBox D510 FreeDos?
FreeDos is itself an OS. One doesn't install another OS on top of FreeDos. But, if you want to install a Linux OS as a dual boot next to FreeDos, there are many distros that will install on an EEEpc. I have Fedora KDE installed on my EEEpc, which, isn't exactly a lightweight distro and it works well. I like the KDE Plasma Netbook setup for the EEEpc.
Any KDE distro that comes with the Plasma Netbook option is a good choice, at least, in my opinion. As I remember Kubuntu is another choice, OpenSuse another. There's also Debian, Mageia, Mepis, PCLinuxOS and siduction. And those are but a few, since there are even more lightweight distros that will probably work well.
You can search for any of these at DistroWatch: http://distrowatch.com/index.php?dataspan=4
My advice would be to download a LiveCD iso of the ones that look good to you, and, place the iso on a usb stick (I bought a removable CD player for mine, if you have one you can just burn the iso to a CD), boot it up live and play with it till you find one that you prefer.