Open the LCM1602 part .fzp , at the top of the file you will have a line of code that looks something like this;
<module fritzingVersion="0.1.1201" moduleId="RioRand_LM2596_94ce9810d0922bc731e2f315085d46ab_1" referenceFile="resistor.fzp">
The "moduleId="RioRand_LM2596_94ce9810d0922bc731e2f315085d46ab_1" is your unique module Id. Make sure you have a "moduleId" in the .fzp file. It can be just about anything you like as long as it matches the moduleRef in your Fritzing/bins/my_parts.fzb file and is not identical to another unique module id or moduleRef... It must be unique...
You can open my_parts.fzb in a text/xml editor. You will find a block of code that looks something like this, with the same .fzp name in the address...
<instance moduleIdRef="RioRand_LM2596_94ce9810d0922bc731e2f315085d46ab_1" modelIndex="85708358" path="C:/Users/Stephen/Documents/Fritzing/parts/user/RioRand_LM2596_94ce9810d0922bc731e2f315085d46ab_1.fzp">
You can copy one of the module IDs and past it into the other or just make up one and copy it into both. As long as they are the same and do not duplicate another part it should work...
After you enter the correct unique IDs into both files and save them. close Fritzing and restart it, Your little warning box should not appear...