Who Counts the Days, and why?

Solution 1:

The simple, boring answer is "we don't know for certain." The longer answer contains lots of speculation and is spoilery as Hell. Like. Seriously. There are so many spoilers in this next bit. If you don't already understand what the phrase you mentioned means, and are hoping to work it out for yourself, then please, don't read it.

So there are a few things that I recognize about that phrase you dropped right off the bat. The "bargain" prrrrrobably refers to the deal the Bazaar made with the Judgments. While the Bargain holds, the Bazaar, the Masters, and the Neath are safe from the tender ministrations of the Dragons. The Name is the name of Mr Eaten, who was given to the priests of the Third City as a sort of combined sacrifice and meal (they became the God Eaters). Mr Eaten is still around, albeit incorporeal and absolutely terrifying, and he seeks to bring about a reckoning. The Masters and the Bazaar both want him to be forgotten, forever, and they've erased his 'real' name from history. By Seeking Mr Eaten's Name, you are helping bring about his reckoning, so if the Name is gone, the Bazaar is, once again, safe. I'm not very good at explaining what happened between the Bazaar and the Sun, but I'll do my best. I mentioned the Messengers, earlier; the Bazaar was one of those, but it fell in love with the Sun. All life is governed by something called the Great Chain. All life forms have a place on the Great Chain. The Judgments, or stars, are at the top of that Chain; inter-chain breeding is strictly forbidden, and the Bazaar had a daughter by the Sun. So both the Bazaar and the Sun were sentenced to death via the Dragons, who are sort of the enforcers of the universe. The Bazaar was given a stay of execution (some say it was the now-dead god/Dragon known as Storm who gave it) in order to prepare a defence. And since the Neath is relatively safe from the Light of Law (which is why sunlight kills you once you've stayed down there too long, unless you've imbibed Hespiridean Cider from the Mountain (who is the daughter of the Bazaar and Sun)), that's where it went to start making its case. It has time to steal seven cities, in the hope that one of those seven will have a love story stupendous enough to convince the Judgments and the Dragons that the laws should be bent. So basically, Counting The Days might just refer to "counting down until the Bazaar runs out of time and is eaten by space dragons."

Sorry about that.

Solution 2:

Snowskeeper's answer is very good and likely true. Here is my presentation of what is perhaps an alternative interpretation:

In Sunless Sea, one is now able to collect the letters of the "Name-which-Burns", which belonged to Salt (a being of unclear nature), an agent of the White, who is a Judgement.
Salt was sent into the Neath on a task of unspecified nature, but shed its name and travelled out into the East, presumably abandoning its purpose. This is very speculative, but perhaps Salt was sent by White on some kind of quest against the Bazaar or the bargain?
"The Name" being gone could be a reference to Salt's abandonment of its quest, resulting in the bargain holding. "Counting the Days" could be a reference to waiting until the "TRAVELLER RETURNS" (to paraphrase Frostfound) and Salt completes its task, perhaps damaging the bargain. However, in the Chapel of Lights, one of the final lessons contains the excerpt "You who remain, you know that there is no betrayal.
The Drowned Man [Mr Eaten] was torn that he might feed us. The White comes to fulfill the frozen law. The seventh city will never fall, and all of us will live."
Who knows what this implies for the actual nature of the White's (and therefore Salt's) purpose? I'm not clever enough to interpret it.