Yes, the position of Arbiter is an elected one. From what Ryza knows, after an Arbiter dies, the True Stone is given to the Guardian, who is an old, absolutely trustworthy manakete who has forsworn all connections to their tribe, as well as the right of ever becoming Arbiter, but are trusted to safeguard the Stone until a new Arbiter is elected.
From there, the various heads of the tribes converge on Ascension Peak and hash out who's going to be the next Arbiter. In theory, any manakete could be elected, but in practice it's usually a Lord/Lady (a leader of a tribe, or someone who has done something extremely impressive). Ryza would remember the Arbiter who was around when she was alive: Vokunfron, who was described by her aunt's letter as someone who's "young, but steady, rational, and willing to listen to everyone before making a judgement."
Nobody quite knows what happens after an Arbiter is elected: they're given the True Stone and there's some sort of ritual that the Arbiter seems to instinctively know, but they never speak of it. They then emerge from the ritual as the Arbiter, but Ryza's reading was a little vague on what all that entails.
