Hard to tell from your description without seeing it. Is the button named in the inspector? Or is it the text on the button you are trying to change?
I assume this a button attached to a script. Instead of clicking on the button, try dragging a box around the whole button in layout mode, to highlight the entire button. If it's in a portal, click on button and drag it out of the portal, then drag a box around entire button.
Right click on button scroll down to 'Arrange' (or click on 'Arrange' menu), and see if the button is grouped. If so, Ungroup button (you may have to do this more then once). If it was grouped you will see multiple light blue lines highlighting all the parts. Click outside button to de-select, then click back on the part you want to change.
If I'm not even in the ballpark, show some screen shots.
A number of layouts have some clever but not always obvious design elements--such as slide controls inside of popovers or transparent objects placed on top of graphics. And there's no documentation included in the starter solution to tell you what was done.
So you'll need to specifically identify the file, the layout and the version of Filemaker that you are using so that others can be sure that they are looking at exactly the same thing that you are looking at in the starter solution.