What lightweight Linux distribution would work well for an old 2 GB netbook?

Lubuntu is probably the easiest lightweight option that should work without much tweaking. antiX is even lighter if performance is the top priority, while Q4OS is already fairly efficient. A clean 32-bit installation may be worth trying if the processor supports it and the hardware has limited resources.

The storage device may matter more than the distribution. Replacing an old mechanical hard drive with an inexpensive SSD can make the whole system feel much faster. If you reinstall, consider wiping the existing Windows/Linux arrangement, giving the new system the available space, and adding some swap space. Make sure anything important is backed up first.

You don’t necessarily need a Dropbox desktop client. The web version should let you sign in and upload or download writing files from a lightweight browser. That keeps the system simpler and avoids using memory on a synchronization service.

Raspbian is intended for ARM-based Raspberry Pi hardware, so it generally isn’t the right choice for a typical x86 netbook. Debian itself works well on older machines if you install a lightweight desktop environment or window manager. Lubuntu, antiX, MX Linux, Xubuntu with Xfce, and Puppy Linux are all reasonable options. For an especially minimal writing machine, Tiny Core or Alpine could work, although they require more setup.