I believe your wish is already granted (with one hitch).
If I do this:
- name: "testing_{{ now().ctime()[:3] }}"
unique_id: 'just_a_test'
state: "{{ now().ctime() }}"
I get this:
However the hitch is I have lost control of the entity’s object_id. It has been automatically set to template_just_a_test (a concatenation of “template_” and the unique_id).
To regain control of it, I can modify it via the UI (not via its configuration).
From this:
to this:
and now it appears with my preferred object_id and templated friendly_name.
The only thing that remains to be seen is if its friendly_name will change to “testing_Thu” tomorrow. 
EDIT
Confirmed; it works (testing_Thu).
