Lightweight, highly portable, good battery life and cheap would be a great combination.
A raspberry pi combined with a battery pack and a case would work out for these - except you'd need a screen, mouse and keyboard. Depends how portable you need it to be I suppose. The Pi would set you back about $30 and depending on the other addons you would probably spend about $100 all up.
Alternatively you could buy a cheap second hand notebook, but the battery life probably wouldn't be that great.