Precious (KJV) is used in the sense of "rare." Rare things are usually precious or valuable.
The sense of this is that the priesthood at that time spoke without inspiration; there was "no open vision." Their messages carried no moral authority because God was not with them. Obviously, Eli was not a very good priest, and his sons were even worse. They did not make the truth open or clear to the people; they were not hearing the inspired Word of God. The people were no longer positively affected by the ceremonies being performed by a decadent priesthood, so through Samuel, God raised up a new moral power to correct the situation.
There does not seem to be a systemized process of succession from one prophet to another. Each prophet received his office directly from God by appointment. This is another distinction between a true prophet of God and a priest, even if a priest speaks under the inspiration of God. A prophet was directly appointed by God, whereas a priest received his office simply because he was a descendant of Aaron.
The classic prophet was a man who preached the way of God to the Israelites yet tended to be outside the established system. This becomes clear from Samuel on.