For example, this statue here. Was that formerly something else (a javelin, a staff, a sword, a torch)? Or is that really supposed to be a small stick?
What is it and what is its significance?
Most likely it was a spear. She was a goddess of war and often depicted with shield, spear and helm.
In literature Athena was often described as holding a spear:
But Athena sprang quickly from the immortal head and stood before Zeus who holds the aegis, shaking a sharp spear…
Homeric Hymn 39 to Athena
[The artist] gives her a shield, she gives a spear sharp-tipped, she gives a helmet for her head; the aegis guards her breast.
Ovid, Metamorphoses 6.70
Her head was covered with a gleaming helmet which was itself crowned with an olive-wreath; she bore a shield and brandished a spear, simulating the goddess' fighting role…
Apuleius, Golden Ass, 10.33
In the visual arts she often appeared with a spear on vases, mosaics and statues.
You can check out this fascinating page for more information and images.