Thanks so much for the information and the investigation on this (I saw you posting in other forums as well). Through some trial and error I think things are working finally. Here is the story for anyone else that might suffer from the same problem.
I originally tried the SIM trick and took the SIM out, put it into another phone, added a contact, and then put the SIM back in. This unfortunately did not work. A few hours later the icons had moved again. I then noticed that the contact that I had added to the SIM was not in my iphone address book, so I imported the contacts from the SIM (Settings>Mail, Contact, Calendars>Import Sim Contact), hoping that this would fix the problem. That didn't work either. I turned the phone off and on again, and that didn't work either.
Then I began to look into the "Claro Contactos" issue, and went looking for the SIM Applications (Settings>Phone). Now, I am almost positive that when I looked here earlier that day the SIM Applications button was not there. Maybe it was, but I'm thinking that by putting the chip into another phone and then putting it back into the iphone, this "unlocked" this feature. Or maybe it was the simple act of taking it out and putting it back in. Either way it was there now. I had never heard (or turned on) Claro Contactos so I went hunting for where to turn it off. After finding it "Utilidades Claro" it popped up a message asking if I wanted to turn it on. Obviously I had not used the feature before so I couldn't turn it off like the previous post had suggested. So I did the opposite and turned it on. I then touched the button "Respaldo Contactos" to back up that one contact to this service and then touched the button "Respaldo Automatico" and told it NOT to back up automatically. After doing all of this, I turned the phone off and on again, and now everything has been fine. It's been a full day and my icons are right where they should be.
I mention the process because I'm not totally convinced that just by turning the Claro Contactos on was the issue. I think the whole process is the solution. Maybe someone else that has the same issue can try the process, but backwards and see if just switching the Claro Contactos on (or off) solves the problem.
Thanks for the help and I hope this helps whoever else might have the same problem.