On a related question. Which is better: the reboot or the remake?
There have definitely been some creditable movie remakes over the years, but I can't think of a decent TV show that came back better than the original with a whole new cast/theme.
The reboot has the advantage of bringing everyone back who made it a success in the forst place, but it's definitely a challenge for it not to feel forced.
I also am a bit of a sucker for the reunion show (Return to Gilligan's Island anyone??)
I definitely prefer the continuation. Dr. Who has a pretty cool concept, though.
In terms of remakes of TV shows that were watchable... Battlestar Galactica? That's not counting shows that were remade after being in other countries, like The Office and House of Cards.
I was gonna mention Battlestar Galactica earlier in the thread until I re-read your guidelines, but certainly one of the few examples of not only success, but a quality one.
Otherwise, I hate remakes. You mentioned He-Man earlier, one cartoon that was remade that I liked was Thundercats in 2011. I loved that show, but cancelled. If anyone can watch the 4th episode, Song of the Petalars, without crying he has no soul.
Another continuation series that I think hasn't been mention was Veronica Mars. Had a film 6 years after and a new season 12 years after. Haven't seen this latest season though.
Wasn't 5 years, but 4, but we did get that Serenity movie that follows Firefly.
Then there's Psych and the follow up movies.