Rumors
Rumor #1: Is Gourry related to Rowdy from the movie, Slayers Perfect (aka Slayers: The Motion Picture)?
The rumor is that Gourry could be related to Rowdy from the Slayers movie. In the movie, Rowdy asks Lina to save the island of Mipross from a mazoku by going back in time and defeating him then. He also wanted her to save Meliroon, an elf girl he loved. Rowdy had the Sword of Light which, among other things, leads people to believe that he and Gourry are related.
Evidence of Truth:
This isn't really a rumor: it is true. Gourry Gabriev and Rowdy Gabriev have the same last name. Also, as mentioned before, Rowdy had the Sword of Light, which is a family treasure passed down through generations, so it explains why Gourry has it. So unless Gourry or one of his ancestors found it and claimed that it was a family treasure, Gourry and Rowdy would have to be related. Also, Gourry and Rowdy look somewhat alike.
It is also believed that Gourry is related to Meliroon, which makes him part elf. But according to Kanzaka, Gourry is not actually related to Meliroon. (This information provided by Ameria.)
This isn't actually a rumor, but merely a fact that isn't directly stated in the anime.
Rumor #2: Is Naga really Amelia's sister?
This is a very popular rumor that doesn't have as much evidence as the previous one, but is widely regarded as being true. Many people think that Naga is Amelia's sister, Gracia, who ran away after Amelia's mother died. Though there is nothing really obvious, there are several clues that lead one to believe that Naga and Amelia could be related.
Evidence of Truth:
In episode 17 of the first season, a woman dressed like Naga, only with red hair walks by and Amelia looked at her as if the woman were familiar. Also, in TRY, Gourry mentions that Amelia had grown, and she says, "My sister was about your height." Naga and Gourry are about the same height. Finally, in TRY, when Lina laughed just like Naga, Amelia said, "I hate that kind of laugh," which could imply that she is talking about the way Naga always laughs.
Rumor #3: Rezo is Zelgadis' grandfather and great-grandfather.
This one has confused many fans and caused them to ask, "How on Earth can Rezo be a grandfather and a great-grandfather to Zel at the same time?!" In episode six of the first season, Zelgadis said, "I'm not sure how old he is, but I imagine he is my grandfather and great-grandfather." But is he, really?
To set the record straight:
If the previous quote made no sense to you, try replacing "and" with "or". The source of this rumor is a mistranslation. Zelgadis said that he isn't sure how old Rezo is, so he is either his grandfather or great-grandfather, but Zelgadis doesn't know which. According to Hajime Kanzaka, Rezo is actually Zel's great-grandfather.