Okay, so The X-Files is one of my all-time favorite shows. I was literally just daydreaming a second ago about how I wish the series could be rebooted. This idea just popped into my head. I don't mean to toot my horn, but I think it's an awesome idea!
My Idea For A Reboot Of The X-Files
Setting: The X-Files Universe
Time: Each new episode takes place in the same time that the original X-File episodes aired. Ex- S1E1 is occurring at the same time S1E1 of the old series.
Characters: Two new agents who worked with the FBI at the same time as Mulder & Scully and who do their own X-file-like investigations. I would start off with one male and one female agent, with extra characters thrown in as the series progresses. They could even make it so that these two new characters were officially given their designation with the X-files if the cards are played right, like say there is a deeper conspiracy that even Mulder & Scully were kept out of the loop on.
What Ties The New Series To The Old Series: Footage of each old episode is spliced into the new episode to tie the two series together. For example, the new agents may walk down the hall and pass Scully as she is heading to Mulder's office. Additional footage would be provided so that the audience understands what Mulder & Scully are doing per their original episodes and how the new characters fit into it, or how they verge off on their own stories.
So it's like a branching off or expanding upon of the original series, giving the audience a greater view into the X-Files universe with additional stories and even some of the same stories where the new characters acted in ways to help progress the original series!