In most cases, when peole's names are mentioned in Tana"ch only their and their father's names are mentioned (e.g. - Yehoshu'a bin Nun). In some cases, the grandfather is also mentioned (e.g. - Betzalel ben Uri ben Chur) and in only 1 case I can thnk of (are there others?) the great-grandfather is also mentioned (Korach ben Yitzhar ben K'hat ben Levi). Is there a specific reason why only certain people's names go beyond the father?
As you mentioned yourself, there are "specific reason[s] why only certain people's names go beyond the father". However, I don't think there is a hard and fast rule for each example, rather each one is brought for a specific reason.
With regards to one of your examples:
Korach's genealogy was listed in the beginning of Leviticus 16:1 only up to Levi, since Yaakov (one generation farther) specifically requested that his name not be associated with the quarrel he caused. (Rashi)
I also seem to remember a reason for listing Betzalel's genealogy (possibly due to Chur being killed for trying to prevent the Golden Calf), but I don't remember the source offhand.