If you're not gaming, then all add-in cards are going to be overkill and have boatloads more memory than you need. Integrated graphics w/ a 128MB share of your system memory pool would be plenty for Aero and general use. Tests have shown that more than the offically recommended 128MB doesn't provide much benefit, if any, for general desktop use w/ Aero enabled.
Frankly it depends on the OS that you will be using and your screen resolution...
XP works fine with just 32MB of video ram.
Vista and 7 require a bit higher 64-128MB for smooth operation of OS with some fancy graphics like aero
Not sure about Windows 8-never used it
Most Linux Window Managers would be content with even less video ram than Windows....
Since you will not be gaming the onboard/IGPU should be more than enough for you.
If you can get your hands on an older video card for free then use that. But otherwise it is not worth getting a video card for just general web/office computer use. Even a $25 Rasberry Pi computer can handle 1080p video output over HDMI.
For my parents I just put my outdated nVidia 7600 video card in their desktop since I did not need it anymore. Even today that video card would still be good enough for what they do and get video processing off of the CPU and system memory.