You cannot put a checkbox into the usual <select>
or multi-select HTML element.
However, here is another question where several good options are discussed.
This looks like the most useful and best suited to your purpose:
https://stackoverflow.com/a/27547021/1447509
And here is an example of how to change the default checkmark to an X:
https://stackoverflow.com/a/40123793/1447509
Sources:
How to use Checkbox inside Select Option
After selecting check box Instead of Tick symbol need X in html
UPDATE:
Given that you need both the HTML markup and the javascript to make it do what you want, you have two (possibly 3) steps to do:
This answer provides a good example of how to create the custom-rolled <select>
control.
This answer shows you how to replace the checkbox created in step 1 with an icon/image of your choosing.
The javascript to remove the x'd <option>
is very simple:
$(this).closest('option').remove();
- IF you also need to save these results, then you also need to learn:
4a. Server-side SESSIONS (so that each user's customizations are saved for them)
4b. A login system, so you know for which user to save the current customizations.
4c. Just the basics of how to use a back-end database, such as MySQL/MariaDB, in which to store the user customizations.
4d. AJAX - so you can schlep info to the back-end for insertion into the database without refreshing (or navigating away from) the current page. AJAX replaces the ancient and no-longer-used <form>
construct. Frankly, once you've used AJAX a couple of times, you'll never go back. Totally easy.
If you are in a bind and need someone to create the whole thing for you, I refer you to one of these websites - I have used such services myself and can recommend them.