How to edit user attributes in AWS Cognito User Pool for specific user?

The console does not have that capability but it can be done in the AWS Command Line Interface.

aws cognito-idp admin-update-user-attributes \
    --user-pool-id xxx \
    --username yyy \
    --user-attributes Name=xxx,Value=yyy Name=ttt,Value=sss ...

Custom attributes use the following syntax:

--user-attributes Name="custom:attributeName",Value="value with space"

You need to set up your credentials with cli first (only the first time), via the command:

aws configure

but after that it is straightforward. For more:

aws cognito-idp admin-update-user-attributes help

The new Cognito console interface is able to edit now.

From console go to Amazon Cognito > User pools > {the user pool you want to select} > {user you want to edit}

Under "User attributes" card, click "Edit" on the right

The OP asked how to change user attributes in Cognito. While the primary User Attributes and the custom attributes, can he viewed in the console, they cannot be changed there.
Here is a link to the Python API to access User properties: link. The update_user_attributes() method may be what you are looking for.

A broader set of APIs for cognito can be found here.