What is the opposite of the Bandwagon Effect? Is there a name denoting the psychological bias where a person prefers obscure methods, things, opinions, etc. over a more popular alternative?
The term "bandwagon effect" was originally used to explain voting behavior. In 1950, Leibenstein introduced it as a concept in economics to describe changes in consumer demand: 


  By bandwagon effect we refer to the extent to which the demand for a commodity is increased due to the fact that others are also consuming the 
  same commodity. It represents the desire of people to purchase a commodity in order to get into "the swim of things"; in order to conform with the people they wish to be associated with; in order to be fashionable or stylish; or, in order to appear to be "one of the boys." 


He also defined a term that describes the opposite pattern of behavior, the "snob effect": 


  By the snob effect we refer to the extent to which the demand for a consumers' good is decreased owing to the fact that others are also consuming the same commodity (or that others are increasing their consumption of that commodity). This represents the desire of people to be exclusive; to be different; to dissociate themselves from the "common herd." 


These definitions have coined the usage of the terms "bandwagon effect" and "snob effect" in micro-economics.

In psychology, the terms have been less relevant. Social, personality and consumer psychology have used other labels investigating people's diverging responses to social influence, such as their tendency for "conformity", or their motives for seeking "uniqueness" (e.g., Berger &amp; Heath, 2007; Snyder &amp; Fromkin, 1977) and "optimal distinctiveness".

References

Berger, J., &amp; Heath, C. (2007). Where consumers diverge from others: Identity signaling and product domains. Journal of Consumer Research, 34, 121–134.

Leibenstein, Harvey (1950). "Bandwagon, Snob, and Veblen Effects in the Theory of Consumers' Demand". Quarterly Journal of Economics 64 (2): 183–207. doi:10.2307/1882692.

Snyder, C. R., &amp; Fromkin, H. L. (1977). Abnormality as a positive characteristic: The development and validation of a scale measuring need for uniqueness. Journal of Abnormal Psychology, 86, 518–527. doi:10.1037/0021-843X.86.5.518