When you're using a geoprocessing tool and have two layers with the same name in your map, it will assign a numbered suffix to each in order to differentiate the two in your list of input features. In the example below, the two layers named "Test" in the contents pane are listed as "Test:1" and "Test:2" in the input features drop-down list within the geoprocessing pane.
Your model could have a similar naming issue. You can see a list of all your model elements by clicking "Report" on the "ModelBuilder" ribbon while you're editing the model. Hopefully that list will help you narrow down what part of the process is causing the duplicate names so you can edit the name of one and eliminate the numbered suffixes from appearing.