It seems to me that Snape's choice of password for his study when he was Headmaster was kinda weird.
Harry ran without stopping, clutching the crystal flask of Snape's last thoughts, and he did not slow down until he reached the stone gargoyle guarding the Headmaster's office.
"Dumbledore!" said Harry without thinking, because it was he whom he yearned to see, and to his surprise the gargoyle slid aside, revealing the spiral staircase behind.
(Deathly Hallows, Chapter 33, The Prince's Tale).
As I say, this is a bizarre choice of password. To confirm, at this stage Snape's cover was very much intact. To the Death Eaters and the Order alike, he was believed to be a committed Voldemort-supporter. Which makes the figurehead of the anti-Voldemort movement a rather odd choice for your office password. It seems reasonable to suppose that Snape would have hosted meetings with fellow Death Eaters (such as the Callows) in his office during his tenure. Indeed, he probably had Voldemort himself up there at one point.
And now everything was cool and dark: the sun was barely visible over the horizon as he glided alongside Snape, up through the grounds towards the lake.
"I shall join you in the castle shortly," he said in his high, cold voice. "Leave me now."
(Deathly Hallows, Chapter 24, The Wandmaker).
Wouldn't that have been a slightly awkward moment, with Snape having to explain his choice of password to his supposed master? "Hmm, why did you choose Dumbledore - the man whose grave I've just ransacked - as your password, Snape?" "Errr...no reason".
Wouldn't Snape's choice of password have seemed odd to friends and foes alike? Didn't it risk blowing his cover in some way? If so, why did Snape choose it?