The Deaths of Argus and Zorya Vechernyaya
Two of the old gods have died during the course of American Gods season 2 so far. Zorya Vechernyaya, the Slavic spirit of the Evening Star, was brought down by a sniper's bullet during the new god attack on an old gods conclave Mr. Wednesday had organized at a diner in American Gods' season 2 premiere. Two episodes later, in "Munin," Laura Moon brought about the death of Argus, who was once the many-eyed, ever vigilant servant of the Greek goddess Hera but was revamped by Mr. World into the new god of Surveillance, under Technical Boy's control. Laura killed Argus by stabbing him in a tattoo on his neck, which Mr. Wednesday claimed was the source of his power.
It had been shown earlier in "Munin" that Argus had died several times in the past, being reborn into a new form as time went on and his legend changed. This was revealed as Mr. Wednesday and Laura attempted to track Argus through his realm, wandering through the sites of his previous deaths. This suggests that, for the old gods, death is far less permanent a state than it's for the new gods.
Indeed, by considering the circumstances of the deaths we have seen so far, it seems that it's far more difficult to kill an established old god than it's to kill a new one. The deaths of Media and Technical Boy come about through mental trauma and torture. The deaths of Zorya Vechernyaya and Argus, by contrast, are the result of a direct physical attack. While it's entirely possible that all of these characters may be reborn before the end of the series, it's unclear just how much control each god has over this process, with New Media noting that the former Media's voice still lives on inside her head. Mr. Wednesday also spoke of having died once before, sacrificing himself to himself on the World Tree, and the experience only serving to make him stronger.
One possibility to consider is that a god may become stronger by virtue of having a proper name and legends associated with that name. While it seems that gods do require some connection to a force that people believe in, like Argus gaining power as an avatar of the surveillance state, this method of feeding is less effective than direct worship, belief, and knowledge of a god's existence. This may explain why the new gods, who are known only by abstract descriptions of their functions, like Technical Boy, are far weaker in a direct conflict than named gods like Odin or Kali and why the new gods tend to favor sneak attacks and subterfuge over direct confrontations.
Neil Gaiman's Sandman and The Rules Of Godhood
It's worth noting that American Gods isn't Neil Gaiman's first examination of the idea of old gods trying to survive in a modern world. This was also a major theme of Gaiman's first great work, The Sandman - a revolutionary graphic novel series which helped to usher in the modern age of comics and promote the idea of comic books as an art form that could be used to tell complex stories for adults. By looking at Sandman and how it treats its deities, we can glean something of how the rules of godhood work in the similar setting of American Gods.
The plot of Sandman focuses upon Dream - one of seven beings called the Endless, who are personifications of forces like Death and Destiny that sentient beings believe must have an intelligence behind them and, in believing so, create such intelligence. Though god-like in their power and their role in looking after all intelligent life, the Endless differ from regular gods in that they do not require belief to fuel their powers. Ordinary gods in the universe of Sandman have a symbiotic relationship with mortals, providing gifts and protection to their followers in exchange for the regular worship, belief, and/or sacrifices that empower them.
This concept is spelled out in detail in Brief Lives - the seventh book making up the Sandman. It's here that Dream, who is the personification of imagination and stories, embarks upon a quest to locate his missing brother, Destruction. Dream starts his journey by making contact with Pharamond - a former god of travelers and transportation, who settled into a new life overseeing several multinational shipping concerns and a chain of travel agencies. While not as strong as he once was, Pharamond has survived while the rest of his pantheon died off, subsisting on the power offered by tending to people's need to travel and send things from one place to another.
With Pharamond booking his travel arrangements, Dream seeks out several of Destruction's old friends, including Ishtar, the Babylonian goddess of war and sex. Ishtar has not done as well as Pharamond and has been reduced to dancing in a sleazy gentlemen's club, feeding on the lust of her audiences in order to live. At one point Ishtar spells out the life cycle of all gods, whom she says start out as dreams and ultimately return to the realm of dreams once the last person who truly believes in their power dies.
This seems to be a fair representation for how godhood works in American Gods, with new gods such as Technical Boy starting out as little more than the imaginary friends of lone mortals until they find a source of power to draw upon. Old gods, by contrast, lose power as they lose followers, but are still able to maintain some degree of their former glory by sheer virtue of name recognition or a connection to their area of influence, such as Vulcan drawing power from the guns and bullets manufactured in his name. It'll be interesting to see how this idea develops and which dead gods might be reborn as American Gods season 2 continues.