A broken staff is an item that can be repaired into different kinds of staves at a repair bench or higher in the Workshop of a player-owned house. Repairing one gives 15 Crafting experience.
Broken staves can be dug up at the Digsite, or fished up on the Fishing Trawler.
When shown to the archaeological expert, he says, "Look at this... Interesting.,. This appears to belong to a cleric of some kind; certainly not a follower of Saradomin. However, I wonder if there was another civilisation here before the Saradominists."
Rates from Mod Ash.
For an exhaustive list of all known sources for this item, see here
Considering the Grand Exchange values and probabilities of obtaining each staff, repairing a Broken staff will on average yield 262. Thus repairing broken staves is hardly worth it even in large bulks.