Have Windows automatically resize the Name column in File Explorer when it opens?

"Windows Explorer should 'remember' the last used column width.

In other questions you mention that you're using Windows XP, changing the BagMRU size may solve the problem.

By default, Windows is designed to
remember 400 different folder settings
(200 local folders and 200 network
folders, for a total of 400 folders).

Start the Registry Editor
Go to HKEY_CURRENT_USER \ Software \ Microsoft \ Windows \ Shell \
Right-click an empty space in the right pane and select New > DWORD Value
Name the new value BagMRU Size
Double-click the BagMRU Size value, and set the Base to Decimal
Now in the Value data field enter the number of folders for which Windows should remember the settings (I'd suggest 1,000 or more, i'm using 5000)
Go to HKEY_CURRENT_USER \ Software \ Microsoft \ Windows \ ShellNoRoam \
Repeat steps 3-6
Close the registry editor"