Given the spec of the machine and the fact you are using unity, as previously answered I think the shell is your problem and not the underlying system. Depending on use and how comfortable you are with Linux command line I'd suggest looking at Crunchbang Linux, it's still based on Debian as Ubuntu is so the command line is much the same but it uses the Openbox desktop environment which is very lightweight. The only thing that might be an issue at first is the lack of a menu button (right click on desktop for menu).
I've got an IBM X61s which uses Intel X3100 graphics, an Intel Core 2 Duo SV T7300 and which admittedly does have 3Gb of ram (but Crunchbang never uses more than 1Gb in normal use) and it's nice and snappy on that. I've also run it on a Samsung Netbook with a 1.6Ghz Atom processor and it runs well on that.
My wife is a bit of a technophobe but she quite happily uses my X61s for web browsing so your Mum should be able to adapt to it quite easily.