Even with Edit My Dashboard permissions and other similar permissions granted by their role, users do not have the option of editing custom views for addition to their My Dashboard.
This issue only occurs when an existing Dashboard is edited i.e. Edit Dashboard->drag and drop Custom view->results in No edit wheel.
When creating a new dashboard (Add Dashboard), the Custom views has an Edit wheel. After this new dashboard is saved, then same behavior as before - no more Custom views can be added/edited to it.
Add this right to your custom role:
Administer Shared Dashboards.
And the definition is:
• Administer Shared Dashboards
• Lets users manage their own dashboards and the dashboards of other users. Users with this role right can edit an existing dashboard page and can save changes that are visible to other users.
o To create a dashboard, the 'Create a Dashboard' role right is required.
o To assign a dashboard to a menu, the 'Administer Menus' role right is required.