Welcome to Android Central! You probably can't use any of the phone's preinstalled ringtones, since they're stored in a protected system directory. But you can create your own custom notification tones by copying any mp3, wav, or ogg file to the /Notifications directory (using a file manager app, or by connecting the phone to your PC via USB and using Windows Explorer).
Not all phones allow the user to assign a custom tone in the Contacts app for incoming text messages -- it's usually for ringtones. I think Samsung phones let the user do this, but I couldn't find anything on the web about LG phones. One workaround can be helpful for people who text you often. Open the messaging thread for a given contact, tap Menu>People & Options, and look for a notification option there. See if you can change the notification tone -- if you can, it will change for that messaging thread only. You can do this for all of your other existing messaging threads. In this workaround, don't delete the messaging thread, because then if that contact sends you a new text, it will start a new messaging thread, with the default notification tone again.