EverQuotes

Designed for Evernote, EverQuotes pulls a random quote from a random note. Notes are converted to plain text, so hyperlinks and text formatting (such as bullet points) are lost.

The default settings are to look at all notes in the Evernote account and to separate quotes by paragraph.

Both those are customizable. You can choose a specific Evernote Tag to pull from if you just want a subset, e.g. "quotable". You can also choose a specific text separator if you wish.

The tablet/phone screen timeout is overridden while the application is active.