Dan(s),
The "grayed out" layer name property in the first screenshot provided in the initial post indicates that the property is not the same for all the selected legend items. There is nothing "locked". You can still work with the property. Please see Work with a legend item—Layouts | Documentation for more information on working with legend items.
Here is an example. In this layout I have a legend with 3 items, Capital, River and States by Region. Note that the layer name is positioned to the left of the symbol for Capital and River, while the layer name States by Region is above the symbols.
When I click the "Show properties..." button in the Element (Legend) pane the Element pane switches to the "[multiple items]" pane. Note that all the legend items are select in the Contents Pane (left side of the screenshot).
The "gray" box next to Layer name" indicates that the layer name for all the selected elements do not share the same properties. Some could be visible, while others are not. Or, as in this case, the positioning of the text name text is not consistent for all selected items.
If you toggle the check box you can turn off the visibility of the layer name for all items as seen below.
If you toggle the checkbox once again you will see the layer name for all the selected items appear in the same relative position.
If you wish to change properties for a single (or a sub set) of legend items select the item(s) you want to work with from the Contents pane as shown in this example. Here I turned off Layer name and rearrange the label and patch order.
These property changes ONLY affect the selected legend items. If I select ALL the legend items again, after turning Capital layer name off, I see that I get the "gray box" next to layer name indicating a property "mismatch" amonth the selected items. Layer name for Capital item is off, while the layer name for the other elements is on.
Hope this helps,
Tom