I can’t tell from that screen shot what you are referring to as a text field, so I don’t know what you want made wider. Zoom in so we can see the objects you want help with. Tell us how this text field was created.
Found it: Object --->>> Text Frame Options
If you mean the text box with the name in it, where the box isn't large enough to hold the whole name, you can just drag a corner of the box to make it larger. You don't have to use the text frame options unless you want to change the inset.