While you can't directly change the username from within your account, you can create a new account and transfer all of your devices from the old account to the new account.
Using the "Sign Up" button at https://remote.it, create a new account using your desired username and password. Complete steps in the account verification e-mail.
Enable the Advanced "Manage Devices" Menu
Next, log into your old account, then go to your Settings -> Preferences:
Select the checkbox as shown:
Create a contact
You will need to create a "Contact" with the target account e-mail prior to transferring a Device.
Click on "Contacts" in the left-hand menu:
Then, click on "Create Contact":
Fill out the form (the Group setting is optional), then click on "Create Contact":
Select all of your Devices
Now in your original account, under "View Devices", check the box at the top of the column to select ALL of your devices.
Initiate Device Transfer
Then, under the "Actions" pull down menu, choose "Transfer Device".
In this dialog, use the "Pick" button to choose the contact for the new account you just created, check the checkbox as shown, then click on "Transfer".
Look for an e-mail coming into the new account e-mail address,
then click on the "accept invitation" link.
Depending on how many devices you have, it could take awhile. Estimate 1 minute per device to transfer and another few minutes for the UI to update.
If you have a paid account
After transferring devices, you should cancel your subscription at the original account and then add the subscription at your new account.