Altium Rename net label, auto update its connection

Select the net label, right click on it, then select Find Similar Objects at the top of the list.

This window will appear:

Select SAME for both Object Kind and Text. Now tick Select Matching at the bottom of the window and hit OK.

This action will highlight and select all net labels whose name is pin 1.

Press F11 to open up the SCH inspector and change the text from pin 1 to whatever you like.

Done!

You can use "Find Similar Objects" function with every type of object in Altium, even in the PCB Editor.

Ctrl+H is the search and replace dialog shortcut for Altium. Additionally, you can restrict the scope of the search to just net identifiers.

If you want to change 'Pin1' to 'Pin5', just use a simple text replacement.

If your concern is accidentally matching 'Pin11' in your search for 'Pin1', the "Whole word option" of the search (and search/replace) dialog is intended for exactly that purpose.

Alternatively, you can use the 'SCH List' panel to edit multiple items in parallel (right click, -> "Switch to Edit Mode"). It also has facilities for limiting the scope (set it to include only net labels).