Natures are pretty basic stereotypes really, you just need to look at what the word means to intuit the reason behind the stat growth. I'm generally going to assume that Atk and Def represent more physical traits, and Sp Atk and Sp Def are more about one's mental abilities. Speed seems to be variable.
...I'm also going by dictionary definitions, making sweeping generalisations, and calling on stereotypes, so for crying out loud don't bite my head off if I inadvertantly offend you, please.
Hardy - "capable of enduring difficult conditions; robust." - so you would expect someone who is hardy to be able to adapt to any circumstances, hence no overt strengths or weaknesses.
Lonely - "without companions; solitary." - someone who is lonely will need to rely upon themselves and may be slightly more aggressive, yet also sensitive to things that may not affect other people...you would think this would be +Sp Atk and -Sp Def for mental strength and weakness, but hey ho, it is what it is.
Brave - "ready to face and endure danger or pain; showing courage." - pretty obvious one here as to why this is +Atk and -Def when you consider that, really. Who needs defence when you're ready to endure pain, and courage implies a certain confidence in one's abilities, hence +Atk.
Adamant - "refusing to be persuaded or to change one's mind." - a general lack of mental awareness that their way is not the only way is a clear reason for -Sp Atk. But have you ever tried arguing with an Adamant person? It's exhausting. +Atk is pretty fitting because with an adamant personality usually comes a short fuse.
Naughty - "badly behaved; disobedient." - If you look at this as describing a naughty child playing a practical joke, then it makes a certain amount of sense why this is -Sp Def, because naughty children can be easily outsmarted by adults, but can also easily take them by surprise.
Bold - "showing a willingness to take risks; confident and courageous." - take what I said about brave and apply some self-awareness to the concept, which completely changes the approach to things in general. Unlike someone who is brave, who will try to barrel their way through by sheer force, the bold will take the slower, safer course. +Def to weather the damage, -Atk because they know they will eventually win through.
Docile - "ready to accept control or instruction; submissive." - someone who is able to do this will need to be adaptive to whoever they are going to surrender control to, so they have no particular need to excel in any one stat.
Relaxed - "free from tension and anxiety." - if you're relaxed then you feel no particular need to rush, so of course this would be -Spd. +Def is a little harder to explain - perhaps being relaxed allows you to defend more readily against your opponent? Think of it like how easily anime protagonists can block the villain's punches after they've powered up.
Impish - "inclined to do slightly naughty things for fun; mischievous." - again, there's an immaturity to being impish that implies -Sp Def as someone more intelligent can see what you're doing and/or outsmart you. +Def means, I suppose, that you're more ready for things to go wrong?
Lax - "not sufficiently strict, severe, or careful.", "relaxed." - I'm not sure which definition to go with here, so I'm going with two that I think could apply in equal measure - I think the general implication here is that the body is ready, but that the mind is somewhere else. Not being tense allows you to more readily take attacks, but being "lax" is generally used as a negative and in this context it suggests a lack of mental fortitude, so -Sp Def.
Timid - "showing a lack of courage or confidence; easily frightened." - as you said, Timid implies someone who will run, and run fast, at the first sign of trouble. Pretty obvious cause for +Spd -Atk.
Hasty - "done with excessive speed or urgency; hurried." - fools rush in where angels fear to tread, right? +Spd, but there are going to be consequences, and you will not be prepared for them, so -Def.
Serious - "demanding or characterized by careful consideration or application." - upon careful consideration, it is dangerous to put all of your eggs into one basked. It makes sense for the serious not to have one stat they are good or bad at because if they took it seriously they would strive to be at least passable at everything.
Jolly - "happy and cheerful." - this is...a little trickier. The way I'd interpret this is that someone who is jolly would be quick to share that happiness, but may not necessarily take everything into consideration...so basically, optimism for optimism's sake. But this is being written by a pessimist who finds optimism unfounded and borderline poisonous at times, so...perhaps not.
Naive - "showing a lack of experience, wisdom, or judgement." - A naive person may be quick to make decisions that they may not if they had the knowledge necessary to make a better judgement, so +Spd for the ability to make choices, but -Sp Def for not making them in full awareness of all the options or consequences.
Modest - "unassuming in the estimation of one's abilities or achievements." - Someone who is modest isn't going to think that their abilities are anything to boast about. But being aware of your shortcomings and limitations, regardless of whether you're downplaying yourself or not, shows a level of mental awareness. So, -Atk and +Sp Atk.
Mild - "gentle and not easily provoked." - With Mild I think of mental strength and control over one's emotions, hence the +Sp Atk and -Def. You wouldn't really NEED defense because you're not easily provoked...and when you are, god help the one who did.
Quiet - "mild and reserved by nature." - See above. It's the quiet ones you have to watch out for. Slow to anger for whatever reason, but get them angry and you're in for a bad time. This just emphasis the lack of speed instead of the lack of need for defence.
Bashful - reluctant to draw attention to oneself; shy. - so, one who is unassuming and unwilling to draw attention to the fact that they're good at anything in particular. No stats stand out above the others by design.
Rash - acting or done without careful consideration of the possible consequences; impetuous. - A more dangerous version of Hasty: those who are rash are probably aware of the consequences, they just don't give a damn. Just because they didn't carefully consider the consequences doesn't mean they didn't consider them at all.
Calm - "not showing or feeling nervousness, anger, or other strong emotions." - again, having the mental fortitude and control over one's own emotions, but this time with an emphasis on rejecting violence. Anger can lend someone great strength, and if you don't get angry, you obviously won't experience that...and will have no need to.
Gentle - "having or showing a mild, kind, or tender temperament or character." - similar to above, but with a more positive connotation in terms of sharing mental wellbeing. A kind, sensitive character who has no need for or defense against violence.
Sassy - "lively, bold, and full of spirit; cheeky." - This is...a bit of a weird one, really. "Sassy" these days tends to mean "sarcastic" and they say sarcasm is the lowest form of intelligence but the highest form of wit...so perhaps +Sp Def could be an implication of a certain level of mental sharpness. But -Speed? I'm at a bit of a loss with this one, unless this is to imply that sassy people can, on occasion, be caught out by their own cleverness, or that sarcasm is usually in response to something, rather than something you lead with.
Careful - "making sure of avoiding potential danger, mishap, or harm; cautious." - someone who is careful will consider all their options before acting, so +Sp Def makes sense. But -Sp Atk? Well, there is such a thing as being TOO careful.
Quirky - having or characterized by peculiar or unexpected traits or aspects. - the peculiar and unexpected trait of having no standout traits? Quirky can mean anything, so having the Pokemon as a blank slate across all stats so that you can decide what you want their personality growth - or stat growth - to be makes a certain sort of sense.
