Was there any specific reason behind Benjen Stark joining Night's Watch in Game of Thrones?
When the Night's Watch was formed, it was an honour to be part of it. The Starks are a very traditional house and they are the realm closest to the wall. So I think it was expected that if possible one member of the family joins the Watch. Though nothing, that I remember, supporting or opposing is stated in the novels.
Not only do the Starks have close connections to the Night's Watch, it is also the case that the noblemen of the North who do not inherit their house (younger brothers, like Benjen was to Eddard, but also bastards like Jon Snow) take the Black.
Although not required, it isn't uncommon either. No one batted an eye at Jon Snow taking the Black even though he was not required to.
Jeor Mormont, while he did hold his own house, also took the Black after Jorah was eligible (and in Jeor's eyes able) to inherit the house of Mormont. Although it is never really specified as to why Jeor chose to do this.