[Updated 11 August, 2013]
Not all Android devices provide simple ways to update your Google password, especially if it's your primary account (the one you used when you first configured your device). The Samsung Galaxy series of tablets, for example, provides many options for customizing accounts but seems to hide how to change your password. Some suggestions we found seemed to cause us more trouble and did not resolve the problem. Here is the one that worked for us.
Before following our steps below, check that your Settings are correctly set to sync data. Of course, you would find this under Accounts. If it looks like it's out of sync, try to initiate a sync. That may be all you need to do to fix your password. If not, try this.
1. Launch Gmail and wait for a message to appear.
2. If the message is a prompt for your new password, you're probably good to go (see Note 1 below).
3. If the message is a 'No Connection [ Retry ]' or a timeout, check your Notifications. You're looking for an entry regarding a failed login. Tap this and you'll get the password prompt you've been aching to see.
4. If those don't work, try Settings > Applications > Manage Applications > [ Clear Data ]. Do a PowerOff/PowerOn restart. Launch Gmail. Check your Notifications for a failed login entry. Tap on it and enter your new password.
5. We've seen a couple of instances where the new password didn't work. In this case, we changed the password again and were able to get in with the newest password. It's never happened twice in one session, thankfully.
Change your password using a PC browser, which should force the prompt for your new password from your phone.
If you still can't change your password, you may need to do a Factory
!!! PAY ATTENTION !!! Make sure you have a good backup before you
do this one, as it will wipe all your custom settings and personal
If you haven't already set an alternate way to get into your Google account, now's a good time to do it.