Off first hand experience the most common causes of A string rattle I've come across are those caused by an improperly slotted nut and bridge saddles that are not perfectly parallel to their bridge plate. If this is a fender style headstock there are nuances to slotting the A that most whom aproach slotting every nut the same way just simply dont follow.
The angle down to the tuner has to match the natural line of your string down to it exactly, most over angle the living daylights out of it leaving a gap on the non speaking portion that sits behind the front witness point of the nut. This almost always leads to buzzing as that knife edge isn't enough to keep the non speaking length from rattling without a scorched earth fix like an after market retainer that includes the A string. You also shouldn't carve the excess rise of the nut down to the absurd levels I often see where you rely on maybe less than a millimeter of the nuts sidewall to keep the string stabilized, its should be a proper slot that is only wide enough to seat the string gauge of choice on the A properly while bracing its read non speaking section properly so its doesn't buzz and require you to add any after market fixers. Its intended to be a slot that offers the string a warm hug that keeps it happy and quiet, not the massive open groove I see the result of shoddy work where every instrument is aproached in the same manner.