I'm going to stick with what I know and advise Peppermint 3 as a starting point .. I *know* it runs well in 512MB RAM, and is pretty simple to use for most tasks .. it doesn't install a ton of cruft, yet has access to all the the Ubuntu software so you can install as much cruft as you want later
There are smaller, lighter distro's .. but I think once you start getting smaller/lighter than Peppermint, they start becoming incrementally harder for a new user to get to grips with.
Good or bad, Peppermints LXDE interface is very similar to Windows, it's pretty damn small, light, and efficient .. yet isn't limited in the way some of the smaller/lighter distros are.
As I said .. just a starting point .. I can say that when I bought my second Aspire One netbook (1GB RAM) it came with a fresh install of XP yet was IMHO pretty unusable .. my other Aspire One (512MB RAM) was MUCH quicker.
That said .. don't expect a system with a 1GHz Transmeta Crusoe TM5800 processor and 744MB RAM to ever perform like greased lightening no mater what you put on it
"Speed" can be very subjective, depending on perception and what you're testing .. which is why I'm advising you take a clonezilla image of Windows first .. again I can say the AA1 never seemed to finish booting XP, continual hard drive activity and ran like swimming in honey .. Peppermint FULLY booted in around 25 seconds .. but the AA1 has a faster processor than the TC1000.
My point is .. yes expect a useful improvement in performance, but don't expect too much.