Well ~がる can mean to «act as if~» or «behave as if~». I don't know who is acting here, but let's say it's her teacher. In that case my best guess at what the non-passive form of the sentence would be something like,
私は先生が(私を)少し奇妙がる子供だった。
I was a child that the teacher acted as if was a little strange.
However who acted as if she was strange isn't really important, so passive is used here to shift focus onto herself as the recipient of the verb. So it's more about how she was treated by others (or her perceived opinion of herself from others).
私は(先生に)少し奇妙がられる子供だった。
I was treated as a little strange child (by my teacher).
(lit. I was a child that was acted as if a little strange by my teacher)
So in essence people around were showing signs that she was a bit strange when she was a child. So although it might not translate well into English, it should act much like any other passive verb in Japanese.