My story isn't very exciting, but I've never shared it on MN before.
When I was 3 or 4, I went through a period of having night terrors. I would suddenly start screaming like a banshee in the middle of the night, waking my parents with the kind of shockingly piercing cries only a terrified child can make.
One night, my father was tangled in the sheets and ended up falling hard out of bed in his hurry to get to me, barely awake himself. I'm not sure if it was that night or on another occasion, but once when they went to calm me, I'd stopped screaming by the time they got to me.
Instead, I was sitting up in bed, staring at something they couldn't see and not immediately responding to my parents. I was speaking incoherently, looking all the while at nothing, so far as they could tell. My mother asked who I was talking to, and eventually I answered that I was talking to that old woman... and pointed into the room, where there was nothing remotely resembling a person.
This was creepy and interesting enough that they mentioned it in passing to my grandmother and great-great-aunt, who lived with my grandparents for as long back as I can remember. My parents thought it was eerie, but nothing more than a child's nightmare; however, my g-g-aunt matter-of-factly told them that if it happened again, they should stand behind me and look over my shoulder, which would allow them to see what I was seeing. 
Apparently there was some story that the people who lived on our land before we did (in a house that had been torn down in the meantime) may have hidden some money somewhere on the property (buried it, perhaps). My g-g-aunt said that they should get me to ask the ghostly woman (or maybe ask her themselves, when they saw her) where the money was hidden.
As far as I know, I never spoke to my parents about the old woman again, and we never found any hidden money, even when we eventually built a new house on the exact site of the old one.
I'm sure it was just a nightmare, but I can talk myself into getting goosebumps over it, if I try hard enough...
Especially since, when I was older (maybe 9 or 10?) and my family was living in the new house, I had another odd experience. One morning, in the first instant of opening my eyes, I thought I saw an elderly woman (with long, grey, wavy hair flowing back from her face) standing at the foot of my bed, looking down at me. [shocked] I blinked, and she was gone.
Again, I'm positive it was the remnant of a dream or a trick of the light, but still a little creepy...