I edited this answer as it seemed to be incorrect.

When you find personal information in your .blend file (such as folder names), you can delete the entire DATA block that includes that information and the file is likely to still work. Do this in a copy to avoid mistakes.

Using any free hex editor (such as HxDen), open the .blend file and search for the personal information that you found in the .blend file. Now click-drag select starting from and including the "DATA" word that precedes the information that you want to delete, and select all the way until you reach the next "DATA" word. Stop the selection right before the next "DATA", don't include it in the selection. Press delete, save the file, open it in Blender and save it again so that any bad formatting is restored, if there was any.

I edited this answer as it seemed to be incorrect.

When you find personal information in your .blend file (such as folder names), you can delete the entire DATA block that includes that information and the file is likely to still work. Do this in a copy to avoid mistakes.

Using any free hex editor (such as HxDen), open the .blend file and search for the personal information that you found in the .blend file. Personal information is usually around this string: "Save As Blender", without the quotes. Try searching for it. Now click-drag select starting from and including the "DATA" word that precedes the information that you want to delete, and select all the way until you reach the next "DATA" word. Stop the selection right before the next "DATA", don't include it in the selection. Press delete, save the file, open it in Blender and save it again so that any bad formatting is restored, if there was any.

I've also been having problems with this. It seems related to the 'Remap Relative' option in the file saving dialog.

Leaving it unchecked seems better, it will only store in the .blend file the current directory that you're saving it to, rather than previous directory that the file may have been in.

Then if you save it to the root directory (C:\ or D:\ like Scales suggested in their answer), it will only have that in the file, "D:", for example.

I edited this answer as it seemed to be incorrect.

When you find personal information in your .blend file (such as folder names), you can delete the entire DATA block that includes that information and the file is likely to still work. Do this in a copy to avoid mistakes.

Using any free hex editor (such as HxDen), open the .blend file and search for the personal information that you found in the .blend file. Now click-drag select starting from and including the "DATA" word that precedes the information that you want to delete, and select all the way until you reach the next "DATA" word. Stop the selection right before the next "DATA", don't include it in the selection. Press delete, save the file, open it in Blender and save it again so that any bad formatting is restored, if there was any.