There's no particular reason for wanting to change my prefix for my local IP, I'm just wondering if it's possible.
So can I change my local IP from 192.168.x.x to something of my own choosing? I assume I'll have to edit some of the router settings.
You can change your router's IP to whatever the router firmware will allow. I don't think the majority require you to stay within the 192.168.0.0/24 range as others have mentioned - some real cheap routers may behave this way. I believe most just insist that you enter a valid IP and netmask.
When you assign an IP address to the LAN-facing interface of a router, you are also usually deciding the subnet of your home network. So e.g. if you assign the LAN-facing interface of your router the IP address and netmask 10.0.0.1/255.255.0.0, you are usually implicitly saying every device on your LAN is going to have an IP address from 10.0.0.0 to 10.0.255.254 (with 10.0.255.255 being the broadcast address for your network). Just want to make the point that changing a router IP may potentially affect a lot more than your router.
I think it's important to mention that if you do not stay on the private ranges as others have mentioned, you risk having an address on your LAN that's the same as a host out on the external Internet, and may not be able to reach specific websites as a result.
There are three sets of address spaces for private networks (as in, not routed through the public internet) to use.
If you are using a router firmware like DD-WRT, you can use one of these ranges by setting the router's IP address to be in that range (as in, make the routers address 10.0.0.1 or something like that).
I don't know if your average, out of the box home router will allow you to do this, however.
edit: I just checked my Verizon router at home and it also has an option to specify what IP range it gives out, so this option may be more common than I thought.
edit2: I just read your comment, you do not have to pick these ranges, but you would then be assigning your computers addresses that are available to public use. If you were to do this, you could have trouble connecting to any company that actually 'owns' that address range.
For example: in one of the comments, someone mentioned using Boeing'ss private address space. Suppose you picked the range 188.8.131.52-255. If one of the computers is assigned 184.108.40.206, you'd no longer be able to access Boeing's website (probably - it's possible a packet might make it out, but most likely it'd just time out or something).
In other words, if you don't own the IP address range, you probably shouldn't use it.
There are three private IP blocks commonly used for LAN addressing
Check out the widipedia entry for Private Network