These are updated daily or several times a week: the Yahoo! Daily Crosswords, the USA Today Crossword, and the Los Angeles Times Crossword.
Crossword Puzzles & Hangman Games:
Just Crosswords is an easy way to create an online crossword puzzle. No registration is required — just plug in the words and clues and the site hosts it for you.
You can create online hangman games (and do a lot more) at My Language Exchange. Registration (again, free and easy) is required).
Word Duck has a cute version of online hangman for many user-created topics. Once you register for free you can also create a game using your own words.
Study Stack lets you easily create (after your register) flashcards, hangman games, word searches, and many more activities.
Play Kids Games is the latest addition to this list. Teachers can use their own content for a variety of games, including Hangman.
With the Flash Hangman Challenge, you can easily write a phrase, email it to a friend, and it will automatically be turned into a Hangman game that can also be posted on a teacher/student website or blog. No registration is required.
There are several sites that allow you to create crossword puzzles you can print-out for free. They include EdHelper, Puzzle-Maker, and Puzzlefast.
The well-known Puzzlemaker from Discovery Education lets you make a lot of other word games for print-out, but not crossword puzzles.
educaplay looks like a great free tool where you can easily create a ton of different kinds of educational interactives that you can link to or embed in your site.