Request it from your ISP
Your ISP should be able to issue a new IP address. If not, threaten to change provider, that should do it.
Change your router's MAC address
You might know that your LAN issues local IP addresses to devices automatically using DHCP, providing it with a 'lease' that is tied to its MAC Address.
This is also likely how your ISP issues IP addresses from their WAN (the connection between your home network - LAN, and ISP's network - WAN). To get a new IP address, you can change the MAC address of your router which will (hopefully) release your current IP address and force their DHCP server to provide a new IP address.
To do this, navigate to your default gateway on your browser. If you don't know how to navigate there, it's likely:
192.168.1.1 / 192.168.0.1 / 192.168.100.1 / 10.0.0.1
Alternatively, use Command Prompt -> ipconfig. It should appear like this:

Log into your router's admin interface and change MAC address. It's different depending on make/model and may not even be available if you're using an ISP-issued router. For me, it's under Advanced -> MAC Address but may also be under Setup -> Advanced.
If you're using an ISP-issued modem/router THEN using your own router, change the MAC Address for WAN0. Restart your router and wait 5-10 minutes and you should be issued with a new IP address. Note that your connection will likely take longer to restore than a typical router restart, so be patient.
Turn your router off and on
If you can't do the above, try turning your router off and on. Some ISPs have 'stickier' IP addresses, so you'll need to do it for at least an hour. This should hopefully release your current IP address, and when your router is back online, issue a new one. Unfortunately, this isn't 100% reliable and is just up to trial and error.


