For the NRA's law suits in Chicago and San Francisco to be successful, the federal courts will have to incorporate the 2A's constitutional guarantees via the due process clause of the 14th amendment. While the 2A as written is a limitation on federal power, incorporation would prohibit the states from infringing the right to keep and bear arms.
Then, whenever a state did violate the 2A, a citizen could mount a claim under 42 USC Section 1983, which provides for compensatory and punitive damages, attorneys fees, and assessment of costs in favor of the plaintiff.
So for instance, suppose you're arrested for open carry in violation of the 2A. They confiscate your weapon, embarrass you, take you downtown, book you, make you post bond, make you angry, etc. You've been deprived of a few hours of liberty. Your actual damages might only be a few thousand -- not enough to entice a lawyer to take the case. But under Section 1983, you can get your damages, plus punitive damages sufficient to deter future violations, and the attorney can submit a bill to the court and have the defendants pay it, so long as it is determined to be a reasonable amount.
Section 1983 has been used to effect enormous social change in the last half-century in the areas of prison reform and equality in employment. If Heller's ruling is applied to the States, we could see a phenomenal expansion of 2A rights.