Automatically restore Texstudio session on startup

Yes. From the menu bar: Options -> Automatically Restore Session at Next Start. Tested on TeXstudio 2.8.0, Mac OS X.

Update TeXStudio 2.10.8

This newer version (on Mac OS X) has the setting in a different place. From the menu bar, select TeXstudio -> Preferences. In the new window, tick the box Show Advanced Options if not already selected. Then in the General category, under the heading Session, there is the option Restore Previous Session at Startup.

As of version 2.10.8 (tested on Windows, but likely true on other platforms), the option has moved to Options -> Configure TeXstudio -> General -> Session -> Restore Previous Session at Startup

(I appreciate that this is now quite an old question, but the answer has recently changed)