In short, Yes. In an otherwise healthy immune system you'd expect that someone with Lyme disease will eventually test positive with Western Blot.
The "whatever it is that Lyme Disease produces that this Western Blot tests for" is worth discussing in some detail. Lyme Disease is caused by a bacterial infection, transmitted by tick bites. As you guessed, there are different strains of bacteria that can cause the disease. They're all members of the genus Borrelia in the phylum Spirochette. In North America, the disease is caused by Borrelia burgdorferi.
Our bodies will respond to the presence of these bacteria by producing molecules to fight the infection called antibodies. The Western Blot tests for the presence of these antibodies (in this case, Immunoglobulin M and Immunoglobulin G) in our system.
In the early stages of the disease, there may not be enough of the antibodies in circulation to show up on the Western Blot. That's where the false negatives come in. In the later stages of the disease (after 4-8 weeks) the sensitivity and specificity of the test goes up to 99%1.
- Harrison's Principles of Internal Medicine. Nineteenth Edition. page 1152