Missing from the "Wall"
Answer: You had a right to use that siddur even if it did not belong to your synagogue because we may assume that the Western Wall authority would be pleased to have their property used for this mitzvah. As regards returning it to the rightful owner, you should check with the synagogue trustees how such a siddur got to where it is. If it turns out that someone mistakenly brought it there after a visit to the Kotel, it is your responsibility as the finder of a lost object to return it to its owner. With so many people regularly visiting the Kotel, it should not be difficult to find a neighbor who will do you the favor of taking it to the Kotel on his next visit.