How to make a custom menu based on a category or tag?

I’d like to have a sidebar widget that has my latests featured posts, which would be marked as a category or tag. Is there a way to do this with the new custom menus, and if not, is there a plug-in that can accomplish this?

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You can absolutely have a sidebar widget that shows posts from a certain category. First, set up your new menu. Click on Menus under Appearances. Then click the plus button towards the top right, and give your new menu a name:
new menu

After you’ve clicked Create Menu, you’ll be taken to a screen to choose what appears in that menu. Towards the bottom left, you’ll see a box labeled categories, with the option to choose from most used or all categories, and to search your categories. Go ahead and choose the category or categories you want to include in this menu, and click Add to Menu (and be sure to click Save Menu at the top right when you’re done!):

choosing categories

Finally, head over the Widgets screen (also under appearances). There, drag the Custom Menu widget over to a sidebar area, give it a title if you’d like and choose your new menu from the drop down. Don’t forget to click save!!
adding widget