Well changing your IP isn't likely from your end and I'll explain why.
When PC's first connect to a network or the internet they don't have an IP at all. They send out a broadcast packet to any/all PC's that can hear it. In that first packet it sends out it's MAC address (a unique number to each network device), and if a DHCP server is on the network it respondes by sending out a broadcast addressed to the original senders MAC address with an IP number for it to use. Something like this:
NewPC: [Broadcast](I have no IP)(my MAC is ABCDEF12)(message:I need an IP!)
OtherPC's: read the packet and ignore it
DHCPserver: reads the packet and responds:
[Broadcast](DHCP IP:192.168.1.1)(To MAC ABCDEF12)(message: You can use IP 192.168.1.55 for 3 days)
Other PC's: ignore the broadcast packet, it's not addressed to thier MAC
NewPC: reads the packet, sees that it's addressed to it's MAC and starts using the new IP
NewPC: [To 192.168.1.1][From: 192.168.1.55](message: Thanks for the IP!)
So what does that have to do with anything? Well the DHCP server uses your MAC address to identify you. If you disconnect from the network and wait more the 3 days(or whatever your lease is), it will take your IP address off the list of assigned IP's and put it back into the pool of available IP's. It might get reasigned after a few more days, it might not. But the way DHCP usually works, if you reconnect a month later it will look at your MAC address, see if your old IP is still available and if it is, you get it back.
So your options are to
A) try and have your ISP flush your MAC from the DHCP table (Very very unlikely they would)
B) buy a new network card
C) try and find out if there is a utitlity to change your MAC address (not really a good thing to play with, but it's the least expensive option)
D) get firewall software to try and block the punks from messing with you.