Is there a hotkey in Visual Studio to open the member drop down list?

Using VS 2008, I still don't know how to get directly to the Members dropdown which is what I want. But, the key bindings will be different for everyone. The command you want to search for under Options > Keyboard > Show commands containing: is Window.MoveToNavigationBar.

Visual Studio 2017+: They now support Resharper-like shortcuts. They work pretty well, too.

• Ctrl+T - 'Go to All' (class/member/file), type m MyProp for members only
• Alt+\ - 'Go to Member in current file'

I think you're looking for Ctrl + F2, which moves the cursor to the navigation bar at the top of a code view. From there, you can press Tab to move the cursor over to the member list.

For this (and pretty much every other Visual Studio hotkey/shortcut key combination), see here.