There are basically two main theories as to why Zephaniah's genealogy goes back four generations:
- In order to connect him to Hezekiah, who is in turn thought to be the reforming king
- In order to establish him as a Judahite despite being the son of someone named Cushi (which might imply he was Cushite)
As to the second idea, the main hole in the theory is the nature of linear genealogies, which are intended to link a person back to some particular individual in the past.1 For example, the genealogy in Matthew 1 is intended to link Jesus back to David and back to Abraham to demonstrate that he fulfills (and is the rightful inheritor of) the promises made to them. Similarly, in Luke 3 the genealogy ties Jesus all the way to being the son of God.2 So here, there must be some point in tying Zephaniah to the figure of Hezekiah, for going back to Amariah would already establish the prophet's Judaic heritage.
The most likely reason is that Hezekiah is the reforming king. Verse 1 goes on to say that Zephaniah is ministering in the time when Josiah was king. As we know from Kings, he too was a reforming king, and so it is fitting that Zephaniah would be his relative of sorts. This perhaps colors the prophecy with the spirit of reform from Hezekiah's day. But beyond that, it probably is not a major hinge on which the interpretation of the prophet's message swings.
1 So R.R. Wilson: "In contrast to the multiple functions of segmented genealogies, linear genealogies have only one: to ground a claim to power, status, rank, office, or inheritance in an earlier ancestor." (1992). Genealogy, Genealogies. In D. N. Freedman (Ed.), The Anchor Yale Bible Dictionary (Vol. 2, p. 931). New York: Doubleday.
2 See also Judges 18:30 where we are informed of the scandal that the priest installed for the Danite idol is none other than Jonathan son of Gershom, the son of Moses.