Point a Domain Name
I have a domain already, and I can add subdomains to it, so I will create a new subdomain, minigames.seanwasere.com
In my domain name provider, I will create a new A name record called minigames for my main domain seanwasere.com that points to the IP address of the new server.
Note that you will be using a domain name that you manage and not seanwasere.com.
Give it some time to propagate, and then visit
http://minigames.your-domain.tld or perhaps you just pointed your main domain to your IP address, and you can then access you games without the subdomain, eg,
Purchasing A Domain
If you don't have a domain name, and your colleagues can't lend you one or create a subdomain for you, then you can get cheap domains from Namecheap
Use the domain name search tool and try anything. The cheapest domains at the moment are the .xyz and .website domains.
Namecheap also has an advanced DNS config tool which allows you to create subdomains. The @ symbol in the image below is an alias for the main domain that is being managed.
The minigames line above, is an example of a subdomain A name record as set up in Namecheap advanced DNS tool for the domain being configured.
The blurred out text in the image above, will be the IP address of your server that your main or sub domain name should point to.