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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.

"Password?"

"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?

I don't think he chose "Dumbledore" as his password. He was an accomplished double agent, and would have chosen a suitable password that would avoid arousing suspicion from either side.

There are a few potential explanations:

It wasn't the password at all

We know from Order of the Phoenix that the gargoyle controlling access to the Headmaster's office has some level of awareness, since it refused entry to Umbridge when she was acting as Headmistress:

"... The Fat Friar told me -" he dropped his voice conspiratorially, so that Harry, Ron and Hermione had to lean closer to him to hear "- that Umbridge tried to get back into his office last night after they'd searched the castle and grounds for him. Couldn't get past the gargoyle. The Head's office has sealed itself against her." Ernie smirked. "Apparently she had a right little tantrum."

_{Order of the Phoenix - Chapter 28, Snape's Worst Memory}

With that established, I think it's considerably more likely that the gargoyle, knowing who Harry was and hearing him call for Dumbledore, simply decided to grant him access to the Head's office.

It was no longer Snape's office

Snape was no longer Headmaster of Hogwarts at that point. There isn't, as far as I know, any mention of a deputy Headmaster or Headmistress during the events of Deathly Hallows. If the position was vacant, it's possible that the interim Head position would be filled by the previous deputy Headmistress, Minerva McGonagall. In that case, the gargoyle may determine a suitable - though temporary password - until she's able to specify one of her own choosing. In that situation "Dumbledore" is as good a choice as any (and far better than most).

It was a password specific for Harry

Due to the gargoyle's at least limited awareness, it's possible that the Headmaster can set a general use password, but also instruct it to open in certain situations or in response to a different password given only by specific people. It's possible that "Dumbledore" only worked because Harry was the one saying it, and that it wouldn't have been accepted as the password if anybody else had said it.

This would, however, require at least some amount of forethought and planning on Snape's part and quite a lot of luck on Harry's (though he's spent the last six years proving he has an uncommon amount of that) for him to get the password without being told it by Snape.

Supposition here:

If it was a password Snape set up, it was easily explainable. It is humiliating a conquered enemy. Stamping his authority on the office. Reminding anyone who heard it who he had defeated to gain access to the office.

As a generally distrustful bunch, Death Eaters wouldn't find it at all odd if Snape whispered the password to the gargoyle in a way they couldn't hear, perhaps using magic to ensure that they definitely didn't hear it.

It's also possible that he simply didn't take anyone there; turning up in their classroom or office would be much more intimidating, which would suit his purposes perfectly - maintain authority through fear, and appear evil to keep his cover intact. As they're all Death Eaters first and teachers second, they may have meetings or conferences as a larger group, in the way that we saw Voldemort holding meetings on several occasions, which would allow Snape to talk to them and give them orders without needing to call them to his office. Having his office remain private and secret would allow Snape to keep an air of mystery; by never inviting anyone in to that private sanctum it reinforces that air of him being above them, aloof and separate.

And of course, as a double agent under considerable stress, Snape would likely appreciate having somewhere safe and private where he could retreat and drop the act for a moment of rest. Using Dumbledore's name as a password is a pretty good way to ensure that no Death Eater would guess it, which could be handy if his cover is ever blown. It's also a small way to honour Dumbledore - after all, it is because of Dumbledore that Snape became headmaster, it was his plan for Snape to take that position after his death - and this fits with Snape's personality in a similar manner to his Patronus taking the form of a doe as a sign of his devotion to Lily.

Another way of looking at it is the password may not have been "Dumbledore".

Albus Dumbledore was nothing if not clever, and Snape was no slouch either. The password may have been no password at all, instead the situation "Harry Potter says a phrase to the Gargoyle when the headmaster is not present".

Whoever chooses a password in Hogwarts may also choose the time frame in which it is valid. For example, in "Harry Potter and the prisonner of Azkaban", the knight Sir Cadogan is using a whole bunch of ephemeral passwords which reduce Neville to despair.

Snape might have set the password for the time after his own death. This would not put him (Snape) in danger as long as he is alive. In this case, his affiliation with the Order of the Phoenix must be known at least by Harry Potter because Dumbledore instructed Snape to inform Harry before he would face Voldemort for the final battle.

This explanation of course requires that it be possible to set passwords on a condition rather than a simple time frame. If this was not possible, Snape may have observed that Nagini was no longer separated from Voldemort which has informed him that the final battle between Harry and Voldemort must be imminent. So Snape might have set the password after noticing that this was the case.