The Prime Minister is the head minister position in Daventry and chief advisor.
Background
It is an advising position to the king like other ministers. However a prime minister may share in ruling of the kingdom when the King is away (though the Seneschel appears to share this duty as well). However, unlike the Seneschel the minister hasn't been shown directly to be in charge of every day castle logistics, finances, or the justice system (though ministers do appear to keep track of castle's supplies, set up parties, and other similar logistics, so as to better advise the king[1]).
Gerwain was the Prime Minister under both Edward, and later King Graham. He stepped down not long after the defeat of the Three-headed Dragon. It is not known who took his position, but may be the Daventry Official another minister of the Castle in later times.
The Prime Minister appears to hold position and duties comparable to that of the Viziers of the Green Isles (though Vizier seems to have more control over the treasury than a prime minister does).
Behind the scenes
It is unclear how much power the prime minister actually has in Daventry. Presumably he would be in charge of the other lesser ministers perhaps. It does seem he was left in charge of the castle while Graham was in Kolyma, but its not 100% clear. In the prologue he seems to be in charge of both castle staff, and setting up parties (a role shared by the Seneschal as well). Historically a prime minister and seneschal are considered the same position as the king's agent, chief councillor or major domus depending on the nation or culture.[2] So this does cause a bit of an issue.
While Gerwain was definitely Prime Minister over the period in which See No Weevil takes place as per the King's Quest Companion, the Seneschel Oswold is left in charge of the castle instead. This could either suggest Gerwain was away, or perhaps the Seneschel is equal to or greater in position to even the Prime Minister. In which case perhaps the Prime Minister becomes an advisor to Oswold in those situations. So this does seem to be a bit of a discrepancy between the novels, and the Companion/games information.
More clues would have to be derived from the court chronicle written by Gerwain during the period of time when Graham was traveling to Kolyma, and the prologue to KQ2, but its not specifically clear that the prime minister was left in charge of the Castle either.