CONFIG_XTENSA_VARIANT_CUSTOM_NAME is not available for thedefaultarchitecture x86.
Result is shown for architecture xtensa

Xtensa Processor Custom Core Variant Name

configname: CONFIG_XTENSA_VARIANT_CUSTOM_NAME

Linux Kernel Configuration
└─>Processor type and features
└─>Xtensa Processor Custom Core Variant Name
In linux kernel since version 3.10 (release Date: 2013-06-30)  
Provide the name of a custom Xtensa processor variant.
This CORENAME selects arch/xtensa/variants/CORENAME.
Don't forget you have to select MMU if you have one.