**Updated 1/14/2017 to reflect that this is done through alexa.amazon.com, which is less obvious today than it was at the time of writing.
***Updated 1/30/2017 to reflect that the Computer wake word has indeed rolled out as the next option. It may have taken a year, but we were 2/2 in predicting the upcoming wake words!
The current state of things
You can check on your current wake word through a browser via the Alexa webapp at alexa.amazon.com. Note that you cannot access this page unless your Echo is plugged in and on a network. The page looks like this:
Here's how you change it to a custom word
*If you plan to do this, please make sure to read the "Bad Results" section below before taking any action.
For the purposes of this example I'm changing my wake word to "Echo". So I edit the value to look like this:
The Good Results of Experimentation
The Bad Results of Experimentation
It is interesting to note that if you have a remote control bound to your Echo you can still issue it commands this way (since you're bypassing the wake word for activation). But it won't do the characteristic "beep" that lets you know it's listening for your input. It just sits spinning in blue mode and will perform any actions you tell it through your remote.
If you get your Echo into a bugged state like this you will also be unable to change the wake word back to anything else. It will give you the error message on everything (even the valid ones "alexa" and "amazon"). The only way we've discovered to get out of this state is to do a factory reset, for which I'll include the instruction here:
To reset your Amazon Echo:
1.) Use a paper clip (or similar tool) to press and hold the Reset button for five seconds. The Reset button is located at the base of your Amazon Echo (near the power adapter port). After you press and hold the Reset button, the light ring on your Amazon Echo turns orange, and then blue.
2.) Wait for the light ring to turn off and on again. The light ring then turns blue and enters setup mode.
3.) Open the Alexa app to connect your device to a Wi-Fi network and register it to your Amazon account. To learn more, go to Set Up Your Amazon Echo.