The Shulchan Aruch (308:8) rules that if the door of a usable utensil falls from the utensil on Shabbat, it does not become Muktzeh, and it may be handled on Shabbat. Even though the door serves no function, and items that serve no purpose on Shabbat have the status of Muktzeh and may not be handled on Shabbat, in this situation the door may be handled.
The principle underlying this Halacha is "Muchan Al Gabei Avihen," which literally means, "It is ready for use on account of its father." Since one intends to reaffix the door to the utensil, it is considered part of the utensil even during the current period, when Halacha forbids reaffixing it. Therefore, just as the utensil which is still usable does not have the status of Muktzeh, the door likewise is not deemed Muktzeh and may be handled.
This Halacha applies to any component of something which Halacha permits handling on Shabbat. If the component falls from its "parent" object, it retains its permissible status and does not become Muktzeh.
One practical example of this Halacha is the case of a screw that falls from a chair or other piece of furniture on Shabbat. Although the screw serves no purpose in its present form, and one is barred by Halacha from returning it to the chair until after Shabbat, it may nevertheless be moved and placed in a drawer for safekeeping. Since the intention is to eventually return it to the chair, it does not attain the status of Muktzeh.
A more common example is the case of a button that falls from a shirt or suit on Shabbat. Here, too, the button serves no purpose in its present form, and Halacha forbids sewing it to the garment until after Shabbat. Nevertheless, since the person intends to sew it after Shabbat, its status is the same as the shirt, and it may be handled on Shabbat.
However, the work "Menuchat Ahava" imposes an important qualification on this Halacha. If the person intends to use the button for a different garment, rather than the one from which it fell, then it indeed becomes Muktzeh. The principle underlying this Halacha is that the intention to return it to the original item renders it Halachically equivalent to that original item. Therefore, in a case where the intention is to return the button to a different item, this provision does not apply, and the button indeed becomes Muktzeh and may not be handled.
Summary: If a screw fell from a piece of furniture on Shabbat, and one intends to screw it back in place after Shabbat, then it is not Muktzeh and one may move it on Shabbat. Likewise, if a button fell from a garment and the person intends to sew it back to that garment after Shabbat, the button does not become Muktzeh and it may be handled on Shabbat. If, however, one intends to affix the button to a different garment, then it is indeed Muktzeh and may not be handled on Shabbat.
Menuhat Ahava, Helek 1, page 284-285.
