Yay. I needed to write up a quick project plan for an upcoming engagement. I refuse to do project plans in Excel on principle.
Installed Project 2013 on my Windows 10 VM. Applied pending Windows update. Created plan. Saved to SharePoint Online. Closed Project.
Went and did something else.
Came back to make adjustments to the plan. Everytime I opened the plan from SP in Project it would open as read-only.
This is a known issue and the KB below will help you.
Ill drop Project 2016 on there tomorrow and see if its O365 integration is any saner. In the meantime the patch below worked around the problem.