Today I want to talk about Eri's hair. Most of it will be family-friendly, but when it comes to matters of hair, there will naturally be a part that is NSFW. I will spoiler these moment the best I can.
Eri has white hair that more than comfortably touches the ground, putting her in an esteemed club with members such as Yukari, Aris and Asuna. Interestingly, Eri's hair is actually on the thin side in terms of students who have their hair touching the floor, since these students typically have their hair like miniature carpets scraping along the ground. Eri, however, has her hair more like a paintbrush gliding across the canvas, concrete or covered, a versatile and trustworthy addition to her arsenal. Something else that is worth noting, though this applies to the entire Occult Research Society more than it does just Eri, is that hair has historically been believed to be magical in nature, potentially containing a portion of a witch's life force, or used to indicate specific targets for a spell. Admittedly, this is information I am not well-versed in, and sources are, rather literally, arcane and fantastical, so this may be somewhat inaccurate. However, I wouldn't be surprised if Eri and the rest of the Occult Research Society take special care of their hair in certain ways, to determine different methods to summon the paranormal. For Eri, she has chosen to have her hair's length be her font.
Eri's ponytail is tied near the end with three red beads, likely meant more to keep her hair in place instead of going hither and thither, getting caught on everything in the process. Naturally, that can't be the only thing that binds her hair together, so she has her hair come together into a small, longer bead-like accessory, giving her hair a much thinner appearance than what may actually be there. I can't imagine Eri would willingly part her ponytail, both to keep her mana tied within the middle part of her ponytail, but also because it would probably be quite a hassle to re-tie it. Perhaps Eri's end is meant to draw latent mana from the ground, her beads being used like a pipe to store said magical energies in her hair, then, when she calls upon the arcane, it gets shot up into the rest of her head, kept in place by her hat, and then given form by her knowledge of the spell. At least, that's one potential theory on how this would work. Due to this being something difficult to truly quantify, there may be multiple ways for her mana to work.
Speaking of, she has a braid on her left side, going down to her collarbone. It's a relatively normal braid, but it has dark red streaks in it. While the amount of arcane energies in a hair are more difficult to get research on, red hair did come up a lot in these studies- that a witch would typically have red hair. This is also where something truly magical occurs- theories on how the red got into her hair. Those of reasoned and formulaic wit would wisely state that she simply dyed or added in the red to her hair. Perhaps. But magick, mysterious melodic motions it may manipulate, could have coalsced into her hair, causing her braid to eventually turn red, especially due to her spells being cast with her left hand- lefthandedness has had a myriad of connotations associated with it, the Latin word "Sinister" meaning "on the left side", which was widely held as the belief of "the other/dark world/side"- essentially being the hand that touches the darkness. Even in East Asia, Chinese alchemists would stir left-handed to draw yin energy, with the Japanese having legends of supposed "kuroi te"- the black hand- able to cast curses. Modern knowledge knows this is due to left-handedness being rare, causing the world to be made for right-handed individuals. As such, it had the benefit of teaching people how to properly and effectively engage in the world. It also potentially had the more sinister advantage of being the hand that wasn't paid much mind, possibly leading to this hand wielding an unseen blade or poison- an early means of sleight of hand. In that sense, is truth really much different from fiction? Perhaps Eri initially put some red in her braid, but it has now since stuck. Interestingly, this appears to be balanced by her earring, which is only on her right ear.
Eri's hat is naturally very important for her magical aptitude- as aforementioned, it can potentially be used to focus her hair's stored mana into her spells, but it also might have use to contain and recieve more magicks. It's possible that the capuccio hat is the cultural grandfather to the modern witch hat, that bred with the sharp tip of the cucullus (hood) of the times, eventually leading to pointed cappuccio being seen as a symbol of a witch, the two eventually becoming merged into one, then romanticized during the European age of Enlightenment, causing the hat to eventually be known as the contemporary witch hat. All this is to say that it's likely that the hat is a really effective way to keep the rain off, which could probably lead to certain individuals pursuing efficiency and rationality to wear such a hat, potentially to the chagrin of others. The hat did have purported magical effects, of course, otherwise it's just a big hat. Effects were purported to be able to protect the spellcaster from their own hexes, being used as a focus that caused spirits to more easily hear the invocations, hiding from surveillance, being a perch for a familiar or, again, being used to give the mana focus by interacting with the caster's mind. The intriguing part with all of these is that they do make some rational sense, either to hide skulduggery, to prevent ill tidings (rain), or a vocal focus (soundwaves getting reflected, likely less effective than the others).
There is also another, simpler reason for the hat, and that would be the rule of cool. Eri likely doesn't brush her hair aside from her long ponytail, kept as clean and smooth as possible to assure that she can more efficiently have mana flow up into her and thus more effectively cast spells. The reason being for her not brushing her hair should be apparent, but it is naturally a very delicate procedure, where if it's done wrong there could be a potential explosion, or she might lose her stored magical energy and have to rebuild it, leading to complications should she need to draw upon the unseen before she has recovered enough. That, and there is the aforementioned inability to cut her hair, due to it being a very important part of her magical energies- while part of it might be because her hair is a bit messy, that's a very superficial look and definitely won't embarrass her for pointing that out. Interestingly, though, the point about hiding things is proven, due to there being a reverse ahoge at the top of her head, going backwards into her hat instead of showing itself. It is possible that this is a very powerful- and dangerous- part of her, even looking at it for too long might drain and bewitch!
Some witch hats have an accessory on them, and while Eri technically doesn't, her halo could be seen as such an accessory. If true, that would mean she has incredible potential, since she may be able to draw upon her own supernatural energies to amplify her spells. This would naturally be extremely dangerous, as it's possible this might be a finite reserve of energy due to it being her consciousness- however, if it were something that could theoretically be developed and strengthened, then that would put her in a unique position as a rare student whose Mystic can actively grow. This becomes a rather wild line of reasoning quick, so I will defer that as a slim possibility. However, since it is still her consciousness, if we do still assume it's possible to use, then that would mean a theoretically powerful enough spell would exhaust her tremendously- it should go without saying that this is not a good thing in the slightest, and should not be encouraged. That, and if she can draw upon her own energies, it's very likely it would only become active in a moment of inconcievable distress. Perhaps it's simply just the aforementioned focus for spirits to hear her request for her aid, though.
Eri's hair is something else entirely. While Arona, Plana and Hatsune Miku are difficult to say how their hair works due to the Shittim Chest being an enigma, where the rules are confusing but the scent is predictable (and Miku's hair seems to imply she might work on the same rules as Arona), Eri is the opposite. Hair in regards to witchery is well-known, though somewhat exaggerated in modern times due to the fantastical nature being inextricable from the historical document- ironically, said fantastical fantasy further fuels the magic, being a potential feedback loop. The issue is not the documentation, but rather the era- the hair of a witch, due to witches typically being documented as commoners, was not really well-cleaned, leading to the scent being the impossible task. The easier part is her hair's cleanliness- it doesn't reflect in the light, but that is because it is shaded. Even when her hat is removed, her hair doesn't reflect, but that would likely be a consequence of this part of her hair being hidden. But it could also be a very clever trick, where she does not reflect light due to the spirits within her hair absorbing said light! Eri's hair would actually fall into multiple categories of hair care- her ponytail, for example, is given the utmost care, not weighed down by any dirt or impurities, likely due to its use in drawing mana from the ground. Her braid is given good care too, but some care has to be put in it- too dirty, and the coalesced magicks will leave. Too vigorous with the cleaning, however, may cause the magick to suddenly disperse, potentially violently. The least cared for, then, is the rest of her hair. Since this has incredible power behind it, while also being where the most movement happens with magical energies, it is inconsistent with how it may react, and is thus very dangerous- a good amount of it may be inert, but there are likely some parts like the reverse ahoge which can't be handled without being dangerous.
As for how her hair smells... The short answer is that it smells magical. The longer answer would be that Eri doesn't want her hair to be smelled, due to the aforementioned magical properties of her hair- it would be very problematic if a spirit travelled along the scent, entering someone's nose! The few whiffs I was able to get carried hints of frankincense, if I'm not mistaken.