1Password offers apps for iOS, Android, OS X, and Windows along with extensions for Chrome, Safari, Opera, and Firefox. The combined apps and extensions allow users to seamlessly access stored passwords with a quick keyboard shortcut. With ease of use being one of the biggest factors in selecting a password manager, it's a shame the Chrome extension doesn't work with 1Password.
The reasoning makes sense, as 1Password integrates with Dropbox and Chrome OS doesn't allow you to directly sync files between Dropbox and your Chromebook. Without the files being stored locally on your device, 1Password won't work.
However, the team at AgileBits has a solution. Granted, the workaround wasn't designed specifically for use on a Chromebook, but it works (although its implementation is bit clunky).
This solution assumes that you sync your 1Password vault between devices using Dropbox. If you use iCloud or manual syncing, I'm afraid this isn't going to work for you.
To access your 1Password vault you'll need to log into your Dropbox account and open the 1Password folder. Next, open the "1Password.agilekeychain" folder followed by "1Password.html".
The last file will open a new tab in Chrome, displaying the 1Password vault lock screen, complete with a prompt asking for your master password. Pin the tab to ensure quick access in the future.
From this point, you can log in, copy, paste, and manage passwords as you would using a dedicated 1Password app. Just remember, in order to actually copy a password you'll need to click on the small "Reveal" button next to a masked password, highlight the text, then copy.
It's not the best solution, but it works.
For more Chromebook how-to content, be sure to check out this page dedicated to all things Chrome OS.