On Magic on Aiming a Spell on Target or Targets discusses how a target becomes willing:
Some spells restrict you to willing targets only. Declaring yourself as a willing target is something that can be done at any time (even if you're flat-footed or it isn't your turn). Unconscious creatures are automatically considered willing, but a character who is conscious but immobile or helpless (such as one who is bound, cowering, grappling, paralyzed, pinned, or stunned) is not automatically willing.
I don't think it's controversial to say that, typically, a creature that opts to make a saving throw then rolls the die then determines the saving throw's result, be that result a natural 1, 2–19 plus the creature's modifiers, or a natural 20.
That said, Magic on Saving Throws on Voluntarily Giving up a Saving Throw only says, "A creature can voluntarily forgo a saving throw and willingly accept a spell's result." This seems to leave but two choices about the meaning of the phrase voluntarily forgo. These are put into context below:
A creature makes a saving throw yet picks a result that fails to willingly accept a spell's result.
A creature doesn't make a saving throw to willingly accept a spell's result.
Such a subtle distinction is rarely needed, but when it is, this GM leans toward #2. That's because the remainder of the game makes it clear—on attack rolls, for instance, and on skill checks—that any attempt the GM allows a typical creature to make has the possibility of success, whether due to a natural 20 or to modifiers that the creature isn't aware of, and whether the creature actually wants to succeed or not.1 And, while a creature can impose upon itself penalties sufficient so that failure is almost certain, the creature's failure—if it makes the attempt at all—is not assured.
The only real way to have no chance of success is not to make the attempt. To paraphrase Yoda, in this GM's campaigns it's try or try not—there is no do and pick the result.
Creature A that's suffering emotional trauma due to its recent dental torture (don't ask) urges Creature B to cast on it the 3rd-level Clr spell heart's ease [conj] (Book of Exalted Deeds 100). Creature B casts the spell, Creature A voluntarily forgoes its saving throw, and the spell's effect ensues.
Ten years later, Creature A decides it no longer needs the effect of that heart's ease spell and urges Creature B to cast on it the 1st-level Clr spell resurgence [conju] (Spell Compendium 174-5), which says, "The subject of a resurgence spell can make a second attempt to save against
an ongoing spell."2 Creature B explains to Creature A that resurgence won't work, Creature A having never made the first saving throw against the spell heart's ease, but Creature B's happy to use an effect like dispel magic instead.
Creature A that's sleeping (therefore equivalent to unconscious therefore willing) is targeted by Creature B's marionette possession spell. The spell has the entry Target: One willing creature and the printed version the entry Saving Throw: Will negates. (The saving throw entry's parenthetical see below is unmentioned by the spell's description.) Creature A is willing so, in this GM's campaign, Creature A makes no saving throw as it's a willingly target of the spell marionette possession.3
Later, Creature C casts on Creature A—still affected by the spell marionette possession—the spell protection from evil, which allows "the subject… [to make] another saving throw (if one was allowed to begin with) against any spells or effects that possess or exercise mental control over the creature." However, this GM would rule that the Creature A—if awake and not willing—is not allowed another saving throw against the spell marionette possession as Creature A wasn't allowed a saving throw to begin with!4
There's an argument saying that, essentially, a creature is in absolute control of its results if those results would typically seem to be less than optimal.5 That is, a GM may allow a Str 10 Medium creature to deal with its dagger 1d3, 1d2, or 1 point of damage; or allow a creature to declare its attack roll a 1; or allow a creature to pick the result of its saving throw—even against poison or disease!—as long as the result is failure. That's absolutely an attractive and fun position, but this GM finds no support for it in the rules.
1 This question discusses voluntary failure specifically.
2 One of my players pointed out the disturbing and darkly amusing duration of the spell heart's ease, which is merely permanent rather than the expected instantaneous. One can, using the spell heart's ease, torture a creature more by first ridding it of the mental anguish caused by its torture then bring flooding back all that mental anguish! There are some messed up spells in the Book of Exalted Deeds!
3 By the way, the spell marionette possession would've failed outright had it been cast on an unwilling creature.
4 The spell marionette possession says first, "You project your soul out of your body and into the body of a willing creature," then immediately after says, "This possession is blocked by protection from evil or a similar ward." The former spell doesn't again reference the latter, making it the GM's option whether the spell protection from evil can have any effect on the spell marionette possession after the spell marionette possession is cast. This author has assumed the example's GM entertained the more upbeat notion.
5 Just an aside: taking 10 and taking 20 are not a check's results but substitutes for die rolls. The check's result is 10, 20, or the die roll plus modifiers, and a creature may be unaware of some modifiers until or unless the GM reveals the result!