RAW for the specter's Incorporeal Movement (MM p 279):
The specter can move through other creatures and objects as if they were difficult terrain. It takes 5 (1d10) force damage if it ends its turn inside an object.
Does this mean the monster could willingly take damage in order to halt inside of another creature or object? If so, would the monster gain a cover bonus?
Note that the Specter's Incorporeal Movement feature does not specify that they take damage if they end their turn inside a creature, only an object.
As for dealing with Objects, yes...a Specter may choose to take damage in order to stay inside of an object and yes, that will give them cover based on how much of them fits inside of it. If they are parked stuck through a table...that's probably half cover. If they are completely inside a wall, that's Total Cover.
As for dealing with Creatures...
The Specter's Incorporeal Movement feature has nothing to say about ending its turn inside a creature. The standard rules for this (on PHB 191) say...
You can move through a non-hostile creature's space. In contrast, you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you. Remember that another creature's space is difficult terrain for you.
Whether a creature is a friend or an enemy, you can't willingly end your move in its space.
As always, specific rules beat general rules, but in this case...there are no specific rules overriding that final statement.
Thus, in the case of dealing with another Creature...a Specter remains bound by the same rules as everyone else because its Feature doesn't say otherwise. The same way that a Halfling can move through the space of a creature that is only one size larger than it, a Specter can move through another creature's space regardless of size. But neither of them can willingly end their turn in another creature's space.
Yes, the creature can willingly end its turn inside an object. Neither the general rules on movement (PHB 191), nor its statbox say anything against that. However:
Whether a creature is a friend or an enemy, you can't willingly end your move in its space. (PHB 191)
What do the rules say about cover?
A target can benefit from cover only when an attack or other effect originates on the opposite side of the cover. [..] A target has half cover if an obstacle blocks at least half of its body. (PHB 196)
First, the attacker and the target has to be on opposite sides of the cover. Although unusual, "inside" and "outside" are opposite sides. Second, the cover has to block (line of effect to) parts of its body. If the object is large enough, that will happen too.
So yes, it will benefit of cover if the object is large enough compared to the creature.
