So, while it's not really important in the grand scheme of things, I was wondering if anyone had an answer for this:

When Bodyguard is discarded to save a character, when does it stop being attached to the character?

Read "What is Moribund?" in the FAQ:

In other words, a Moribund card is considered to have left play but is physically still in play and retains its abilities (including attachments and their abilities) and is allowed to interact with the game as normal until it is finally (and physically) placed in the discard/dead pile or returned to hand/deck during Step 6 of the action window.

This means that any response ability or passive effect on a Moribund card can be triggered for the remaining duration of the action window, and other abilities that trigger when a card leaves play (in either of the four ways) can be triggered (even if the card has not physically left play yet). When the card finally leaves the game in Step 6, it loses its abilities (as cards do when they actually leave play), and ceases to be Moribund. Attachments also lose their abilities (and are discarded).

When Bodyguard is discarded to save a character, when does it stop being attached to the character?

What are you getting at here?

As kauai says, because the discarded attachment is moribund, it follows all the rules for moribund cards. So you can Respond to the attachment leaving play or being placed in the discard pile, but it still counts as an attachment on the character, a card under your control and total attachment counts (for things like Griff) until physically removed from play.

The question I am asking has nothing to do with the ability still working or where moribund goes (or ends). It was specifically: When you discard an attachment from a character during the save window, when does the attachment officially become unattached from the character?

The FAQ says that when a character becomes moribund its attachments are still attached until step 6. It doesn't say anything about an attachment staying attached once it becomes moribund. I have to assume that this is the case, but I can't find anything that says it does, and I have a new-player friend who wants to know.

The moribund attachment being still attached follows from it being still in play. Attachments can't be in play without being attached to something.

I'm just curious whether the question arose out of academic interest or whether there was a practical result you were looking to gain some insight into.

It had to deal with a new player using Pentos, and he just wanted to know when the cards became unattached. Curiosity really.