Scenario: When logging into the UEFI BIOS, you can see the asset tag's current value set as 0, but are unable to change it.

Solution: To change the asset tag on a Surface Pro 3, you must download a utility by Microsoft.

Note: Before beginning, please ensure you are using UEFI firmware version 3.9.150.0 or later

1. Download a copy of the Surface Pro 3 Asset Tag CLI Utility from here (Surface Asset Tag.zip):
   https://www.microsoft.com/en-us/download/details.aspx?id=46703
2. Extract the files from the zipped folder
3. Open up an elevated command prompt
   
4. Navigate to the folder you extracted the Surface Pro 3 Asset Tag CLI utility to
   
5. Execute the following command
   1. AssetTag.exe -s ENTERYOURASSETTAGHERE
      
       

       

      1. Note: The asset tag can be up to 36 characters long. Valid characters include A-Z, a-z, 0-9, period and hyphen.
6. Reboot the machine for the changes to take effect

About the tool

The utility comes with a readme on additional functionality.  This is copied directly from the Surface Pro 3 Asset Tag README.txt file for convience:

This tool gets or sets the proposed Asset Tag, which will be applied on next reboot.

The current Asset Tag is an SMBIOS setting which can be queried via WMI:
(Get-WmiObject -query "Select * from Win32_SystemEnclosure").SMBiosAssetTag

Get proposed asset tag:
AssetTag -g

Clear proposed asset tag:
AssetTag -s

Set proposed asset tag:
AssetTag -s ABc-45.67

Valid values:
The asset tag can be up to 36 characters long.
Valid characters include A-Z, a-z, 0-9, period and hyphen.

PowerShell script demonstrating way to get proposed value and interpret errors.
Note that stout contains the Asset Tag and stderr contains error messages.

AssetTag -g > $asset_tag 2> $error_message
$asset_tag_return_code = $LASTEXITCODE
$asset_tag = $asset_tag.Trim("`r`n")

if ($asset_tag_return_code -eq 0) {
Write-Output ("Good Tag = " + $asset_tag)
} else {
Write-Output (
"Failure: Code = " + $asset_tag_return_code +
"Tag = " + $asset_tag +
"Message = " + $error_message)
}

Update: Microsoft has posted official documentation on these steps here: https://docs.microsoft.com/en-us/surface/assettag