"Changing" a Minecraft server's IP address means different things depending on why you want to do it. Maybe your home internet's public IP keeps changing and breaking connections. Maybe you just want a clean, memorable address instead of a string of numbers. Here is how to actually solve each case.
If your IP keeps changing (self-hosted)
Most home internet connections use a dynamic IP address that your ISP can reassign periodically, especially after a router restart. This is the single most common cause of "my server used to work and now nobody can connect" complaints. You cannot force your ISP to stop doing this without paying for a static IP add-on, but you can work around it with a dynamic DNS (DDNS) service such as No-IP's free plan.
A DDNS service gives you a fixed hostname, like myserver.ddns.example, and runs a small client on your network that automatically updates the hostname to point at your current public IP whenever it changes. Players then connect using the hostname instead of a raw IP, and it keeps working even after your address changes behind the scenes. No-IP also publishes a general port forwarding guide that pairs well with DDNS if you are setting both up at once.
If you want a custom domain
If you own a domain name, you can point a subdomain at your server using an SRV record, which lets players connect using something like play.yourdomain.com without needing to specify a port, even if your server runs on a non-default one. This is purely a DNS change made through your domain registrar, it does not touch the server itself.
If you are moving to a different host
Switching hosting providers, or moving from self-hosting to a hosted server, naturally means a new IP address, since the server is now running on entirely different hardware. In this case there is no way to "keep" the old address; the best you can do is announce the new one clearly to your player base, and, if you use a custom domain, simply repoint the DNS record at the new server instead of changing anything players see. A server status checker is a quick way to confirm the new address is live before you announce it.
If you are on a hosting provider and want a different address
Ask your host. Most providers let you request a different IP or a custom subdomain through the control panel or support, without needing to migrate your world or lose your progress.
Avoiding the problem entirely
The cleanest fix for IP instability is not managing DDNS yourself, it is hosting somewhere that already has a stable, unchanging address by design. Tick Hosting assigns every server, including free plan servers, a fixed address that does not shift around like a home internet connection does. Set one up at Tick Hosting's plans page and stop worrying about IPs altogether.