Organisms as potential Bio-Weapons?
There is both a set "list" of agents, but more importantly, a set of properties that an organism needs to be in order to be truly worrisome.

First, the list:

The CDC classifies agents into one of three categories, Class A, B, or C.

Class A: These are organisms that are hard to control, highly transmissible, and lethal:

Anthrax (Bacillus anthracis)
Botulism (Clostridium botulinum toxin)
Plague (Yersinia pestis)
Smallpox (variola major)
Tularemia (Francisella tularensis)
Viral hemorrhagic fevers (filoviruses [e.g., Ebola, Marburg] and arenaviruses [e.g., Lassa, Machupo])


Class B: These are organisms that are fairly easy to disseminate, make people sick but don't necessarily have high mortality rates, and may be difficult to detect.

Brucellosis (Brucella species)
Epsilon toxin of Clostridium perfringens
Food safety threats (e.g., Salmonella species, Escherichia coli O157:H7, Shigella)
Glanders (Burkholderia mallei)
Melioidosis (Burkholderia pseudomallei)
Psittacosis (Chlamydia psittaci)
Q fever (Coxiella burnetii)
Ricin toxin from Ricinus communis (castor beans)
Staphylococcal enterotoxin B
Typhus fever (Rickettsia prowazekii)
Viral encephalitis (alphaviruses [e.g., Venezuelan equine encephalitis, eastern equine encephalitis, western equine encephalitis])
Water safety threats (e.g., Vibrio cholerae, Cryptosporidium parvum)


And finally, Class C, which are emerging threats that may be dangerous if engineered largely due to their novelty and their potential lethality/transmissability. The CDC uses examples like Nipah virus and hantavirus.

So that's the list of what the CDC thinks is a big deal. You'll note influenza isn't even on there. I'd argue that any given novel influenza strain belongs in Class C, and likely not Class A because, while potentially both transmissible and virulent, a known vaccine exists and simply needs to be formulated.

In terms of properties for an organism, I'd suggest the following, in no particular order (it likely depends on what said organism is meant to do):


Highly virulent: The disease needs to be able to cause human disease, with high rates of both morbidity and mortality.
Highly transmissible: Unless the goal is to cause a "one off" incident, the disease needs to be able to establish a productive chain of human-to-human infections. Humans can't be a dead-end host. Ideally, this needs to be direct human to human transmission, rather than a vector born disease like West Nile.
Stable and amenable to engineering. It needs to be something that's easy to culture and grow in large numbers. This one often gets overlooked somewhat, but despite the fact that its now somewhat commonplace, bioengineering isn't exactly easy.
Not amenable to treatment. The organism needs to not be easily treatable - broad spectrum antibiotics, existing vaccine stock, etc. mean that the normal public health infrastructure can address, or at least mitigate, an attack. For an attack to be truly effective, it needs to swamp existing infrastructure, and one way to do that is to necessitate intensive, unusual care in hospitals and out of the ordinary action on the part of public health officials.