There is no hard and fast "willpower" save or attribute.
The way you define willpower in your question, there is no attribute that exclusively represents "Being able to resist mental effects and emotions effectively, and not letting things deceive you. Also being able to do what needs to be done."There are, however, a few attributes that do not fit this description. Let's go through each attribute and see how they fit into that definition. I am currently without my PHB, but they have some general descriptions of them, which I will add later.
Strength
Generally strength is your physical ability in regards to moving, lifting, pulling, etc, all of which go into general combat and physical contests. This has basically nothing to do with willpower, only what you can and cannot do (and how well you do it).
Dexterity
Dexterity is nimbleness, speed, and precision. This relates to ranged attacks, dodging attacks, and how well you can defend yourself in combat. This also does not relate to willpower.
Constitution
Constitution is like your physical willpower. It allows you to resist poisons as well as maintain concentration. You could argue that maintaining concentration takes a certain aspect of willpower to focus on a spell rather than the pain coming to you. Constitution could directly influence willpower.
Wisdom
Wisdom I think best represents willpower, but again, there is no exclusive ability to this. Wisdom allows you to resist a lot of spells, but also tends to common sense and perception. So wisdom is not willpower, it is something else entirely.
Intelligence
Intelligence relates to your actual brain capacity as well as academic knowledge and logic. It is, in a way, a representation of your physical brain, as it lets you resist psionic effects, tell if something is an illusion, or recall information previously learned.
Charisma
Charisma relates to the emotional aspect you mentioned of willpower. It also relates to your social skills. A person with high charisma can deceive or persuade better, but also resist charm effects. This is another aspect of the willpower of which you spoke
So what does it all mean?
It means that willpower is contextual. When you're looking for a "willpower save" or a contest that isn't outright given to you in the rules, you need to weigh each attribute and how they fit into this particular "willpower save". Willpower against pain seems to fit constitution pretty well, but willpower against lust would more fit charisma (resist that succubus!). Willpower against magic effects is often attributed to wisdom, but willpower against psychic manipulation can fit intelligence as well.
Consider also the technique of allowing your player to decide which attribute to use, as is done in some RPGs. The player picks an attribute and justifies it to you and then they use that attribute. Can they justify Dexterity as a willpower save? I mean, maybe. It's up to them to make the case and you to accept or reject it.
So weigh your options as DM, see which attribute fits the best for your current situation. Or, as a player, weigh which aspects of willpower your character is best at.