So what you are asking is: "How can I do something that the manuals says is impossible?" 🙂
You'd have to setup a new webserver, running under a new and unique userid.
This userid would have create and drop database rights.
On this server you'd put a form for create database and for drop database.
The drop-database list should obviously only offer to drop databases that were created by that same user.
Your shared users would then have password protected access to this form.
Suggestion: force a prefix of the user's login or something at the beginning of the database name, so people don't constantly find out that the database name they planned to use was already taken.