IMO, any reboot needs 2 things: a) the "spirit" of the original, and b) to be able to connect with a modern/foreign market audience. It doesn't have to be the exact same story, but it should strive not to alienate fans of the original. If the reboot is ambiguous on if its a reboot or not, my preference is that it strive to clarify that, rather than be half reboot half continuation *coughHawaii5-Ocough*
Yes, there are some things that don't need a reboot. But there are also a number of things that, if you look back at it now, is incredibly dated, and hard for modern audiences to get into. A good reboot would take that same basic concept, keep the spirit of the original, and modernize it so that a modern audience could enjoy it. Or adapt it for an audience foreign to the original, for a remake of a foreign film/series.
While I personally feel that movies reboot better than TV series (original format only, movies rebooted into TV series seem to work okay), my best example of reboot done right AND wrong are a TV series: Sailormoon.
Sailormoon does have the distinction of being adapted from a source, which always muddies the water on if something is a reboot or just a re-adaptation. In this specific case, we're talking about a source manga, an anime adapted from the source manga, a live action reboot (modernization) of the source manga's first arc, and a re-adaptation of the source manga. But we'll run with it for these purposes.
The original manga and anime of Sailormoon are both incredibly dated. Arcades aren't the big thing they were back when it was first made, 20 years ago. The characters have mini disc players, which a) never really caught on outside of Japan, and b) aren't a thing anymore. The art style is also very "90s."
The tokusatsu Pretty Guardian Sailormoon came about 10 years later. The secret base was now a karaoke parlor, to adjust to the times. CD players replaced mini disc players. There were a lot of changes to the storyline (and one character), and while the feel was much different (PGSM is way darker than anything else in the franchise), IMO, at least, it still captured the "spirit" of the original manga (the anime was much more light-hearted and whimsical than the manga was). I can see why fans of the original anime didn't care for it much, but I think a lot of manga fans accepted it. Focus was placed on developing the characters on both sides, and while there were some... interesting... changes, the changes to the storyline accounted for it.
And then we have Sailormoon Crystal, the most recent outing. Crystal is extremely polarizing in the fanbase. Either you accept it and love it as a "proper" adaptation of the manga, or you loathe it because, quite frankly, it's not very good (I seem to be one of the few in the middle: I see it's flaws, but can at least enjoy it. Also, it got a lot better towards the end, and I'm cautiously optimistic for the next arc). Yes, it follows the source much closer than any other version (to the point of recreating scenes panel-by-panel, line-by-line), but the biggest flaw it had (other than the extremely off-key animation) was that it lacked the "heart" of any of the other versions. The manga, yes, was short and to the point, but you could feel the emotions that the mangaka put into it. The anime, on the other hand, stretched it out to give the characters more development. The tokusatsu took the middle road, giving the characters development that served to drive the plot (interestingly enough, the toku is longer than the first arc of the anime, even though it feels shorter, to me at least). Crystal had none of that. It seemed to have a checklist "this scene, check; next scene, check", and didn't go further (until later in the series).
Of these three, both the original anime and PGSM fit the first criteria, for keeping the "spirit" of the original manga. The original anime had no modernizing to do, since it came out at the same time, but both PGSM and Crystal did fit the criteria for modernizing, with PGSM and it's karaoke parlor, and Crystal adding in laptops (and, yet, still had mini discs and arcades). Both PGSM and Crystal, though, had problems with not alienating the original fanbase (in the former, mainly the original anime's fanbase, with the latter, everyone).