Basically, my question if is it possible to summon a spectator as a bodyguard. But let me first explain it:
Monster Manual p. 31 says, about Spectators:
A spectator is a lesser beholder that is summoned from another plane of existence by a magical ritual […].
Magical Guardians. A summoned spectator guards a location or a treasure of its summoner's choice for 101 years, allowing no creature but its summoner to enter the area or access the item, unless the summoner instructed otherwise. If the item is stolen or destroyed before the years have all passed, a summoned spectator vanishes. It otherwise never abandons its post.
Emphasis not mine.
So, imagine that a character knows that "magical ritual" and summons a spectator. (Note that I'm not asking how to summon it, for the sake of the question the NPC already know and can, or, if it is a player, he knows the proper ritual/spell to do it.)
For example, a wizard from Lost Mine of Phandelver, per p. 48 of the Starter Set adventure book):
The monster that guards this room is a spectator. One of the human wizards who worked in the Forge of Spells summoned the creature to guard the magic items created and stored here.
Imagine that instead of protecting the room the wizard wanted to use this spectator as a guard, could he do that?
In other words, can the treasure or item to guard by the spectator be an alive creature (like the summoner)?
So, the spectator will always follow its summoner to protect him.
In the case that it isn't possible, can the treasure or item to guard by the spectator be a moving object (like a necklace)?
So, instead of guarding the character, the spectator follows the character's necklace to protect it from anyone (less the summoner and friends) who dares to take closer and try to steal it.
Best Answer
Spectators are not bodyguards.
In 5e, the rules do what they say they do (and anything else is up to the DM to decide). The target of a Spectator's attention is "a location or treasure". Its remit is "allowing no creature but its summoner to enter the area or access the item..."
A living creature is not a location, not a treasure, not an area, and not an item.
Even if it were, though, it wouldn't have the effect you want. It doesn't bodyguard, it prevents access. Even if it could be assigned to a person it wouldn't be able to distinguish between stabbing them with a knife and shaking their hand. A bodyguard that can't tell the difference between those two things is not a particularly effective bodyguard.
As far as the necklace, that could work. It's a legitimate treasure, no one's stolen it, and nothing says that a Spectator has to stay in one place. It would certainly help to prevent anyone from stealing the necklace from you, or from taking it off your cooling corpse. On the other hand, it's going to do very little to the part where they convert you into a cooling corpse in the first place. What does it matter to the spectator if someone shoots you full of holes? You're not the one it's guarding.
...and if somehow you convince the spectator that everyone who gets within 20 feet of the necklace is a threat? Well, have fun walking down a city street.
This is not what this tool is for, it is not suited to the purpose, and it won't work.
There are other ways, though.
A spectator is, among other things, a LN creature with int 13, who's capable of communicating via telepathy. If you really want a spectator to help you out, you can talk with the thing, and possibly convince it. The ritual even gives you a way of making one show up and be non-hostile. So, for example, if you're absolutely confident of your ability to bluff the spectator into doing whatever it is that you want it to do in the moment, then binding one to a necklace that you can wear is a pretty reliable way of always having a spectator on hand to bluff.
Alternately, the spell only binds them for 101 years. After they're free of it, they pretty much become free-willed. At the same time, many of them continue to do whatever it was that they were doing before (more or less) out of habit or lingering fondness or some such. If you are particularly long-lived, you might be able to befriend one in the century+ of its servitude, and still have it as a friend when it was done.
None of this will give you guaranteed loyalty or obedience, and it all requires more effort than just "Perform ritual. Receive bodyguard." Also, the Spectator is both an aberration and insane, which is likely to complicate the process. Still, if what you want is a friendly eyeball-critter, it's at least technically possible to befriend an eyeball-critter.
Of course, if you're the DM...
Well, a spectator is summoned and bound through a specific ritual that does a specific thing. It would be entirely reasonable to suppose the existence of a similar-but-distinct ritual that did a similar-but-different thing of summoning a spectator (or something very much like a spectator) as a bodyguard. If that's something that you want to have in your game, it's an entirely reasonable thing to have in your game.