Sol is a mage, but unlike most mages in the game, she can be played well in both the traditional mage role of Mid, but also can work as an ADC.
1Ideal Item Stats
Magical power is a core stat for all damage mages, for obvious reasons. More power=more (potential) damage. To go along with all the magical power you'll build, magical penetration is necessary to get through the base magical protections of your enemies and any magical protections gained elsewhere such as from items in their builds or from relics or a support's auras.
Magical lifesteal isn't a stat that you will focus for it alone, but you'll end up having it by building any items from the Tiny Trinket item tree. Items such as Polynomicon, Book of the Dead, and Bancroft's Talon, have other benefits and appealing features.
Attack speed isn't a necessary stat for Sol but it's helpful to have built, allowing her to output more basic attacks, thus allowing her to deal more damage per second. At most you'll only want two attack speed items- anymore is unnecessary.
Cooldown reduction is a luxury stat on Sol. It isn't one that should be prioritized but is nice to have should you come across it.
Soul Stone is an ideal starter item for Sol. The extra magical power this item grants, gives her stronger early wave clear and jungle camp clear, in addition to stronger poke. The mana return upon consuming the passive stacks is just enough to not have any mana issues in lane. Sands of Time can be an option if you value the cooldown reduction over the extra magical power. Vampiric Shroud isn't a great starter item for mages in comparison to the others- it is difficult to justify building this item over any other starter item. While this may not be a classified as a starter item, it is worth mentioning Talon Trinket. Talon Trinket grants a whopping 60 magical power and 8% magical lifesteal right off the bat, but the appeal of this item is that it allows you to build Bancroft's Talon, a very strong item on Sol, sooner than usual, as you don't have to build into a starter item.
Usually if you don't build into Talon Trinket, you'll want to build into Tiny Trinket, to be able to build into Bancroft's Talon sooner. The stats it grants for that price isn't that bad either. Other options include Lost Artifact which can build into Doom Orb, or Rod of Tahuti in the later stages of the game. Imperial Helmet can also be an option if you would love to build into Dynasty Plate Helm. Bumba's Mask can also be consider for Sol in the mid lane, granting you a bit of extra sustain when clearing camps, and the ability to clear jungle camps faster.
Each god has a few items that you will be building almost all of the time, and Sol is no different.
Shoes of the Magi is an essential item on Sol for obvious reasons. It provides the most offensives stats out of all the items in the Shoes tree. Shoes of Focus aren't really ideal because as mentioned earlier, Sol is not dependent on CDR as her main abilties have a low enough cooldown. Ideally you will build this as your first or second complete tier 3 item.
Bancroft's Talon is an amazing item for mages. Offering 100 magical power for 2300 gold in addition to 20% magical lifesteal plus a passive that grants you an advantage the lower your health drops, and allows for outplays. Sol can make better usage of this item due to the fact that she not only has damaging abilties but her basic attacks also pack a punch- and not only will she be able to lifesteal from her abilites but she will be lifestealing more from basic attacks because she is using them constantly unlike most mages. You can see this item being built as early as a first tier 3 item to as late as a second tier 3 item.
Obsidian Shard is a great penetration item for Sol. This is mainly due to the fact that it boosts her structure damage than any other penetration item. Negating 33% of a tower or phoenix's magical protections is huge. It's usage in combat is good too as the penetration is immediate- you aren't required to stack up protection reduction stacks to obtain the benefits of it. Typically you'll want to complete this as a 3rd item but it can go as late as a 5th item if the situation allows for it.
Another core item is Rod of Tahuti. This item is essential on most mages as it grants a lot of magical power right off the bat, and it also increases your total magical power by 25%. A hidden benefit of this item is that Rod of Tahuti also increases your basic attack damage. Normally your basic attacks will scale from "base damage+20% of your magical power" but with Rod of Tahuti, they will scale from "(base damage*1.25)+20% of your magical power". Traditionally, you will want to build this when you already have 4-5 tier 3 items completed.
Situational & Unfavored Items
These are items that are not quite as important as the aforementioned core items.
Dynasty Plate Helm is a fantastic mage item for what it gives for the price, so it is still a viable option for Sol in general. Usually if you want to build into this item, you'll build it before Bancroft's Talon.
Doom Orb is an alternate choice in place of Bancroft's Talon. This item has the potential to be very cost efficient although it is risky to build as a single death can force you to regain the lost stacks. I would not recommend building this item unless you are very confident in the match.
Book of Thoth in my opinion is not too great of an item on Sol. The only real appeal it has going for it is the high magical power it grants in comparison with other mage items, but that upside is offset by its higher gold cost. Sol is not a goddess that will drain your mana unless you are playing recklessly and have no understanding of ability and mana management.
Soul Reaver can be an okay item but there's the fact that it's a pretty costly item but can be pretty great for Sol Mid or burst oriented builds.
Warlock Sash has great stats...if you fully stack it. Honestly, there are better options available instead of Warlock Sash. Items that are effective much sooner. If you find yourself for some reason, really needing the health this item provides, please consider improving your positioning and awareness instead of relying on an item that isn't even that great.
Ethereal Staff can be an option for burst oriented or mid builds due to its stats and passive. Typically you'll want to build this somewhere somewhere as a 3rd to 5th item.
There is no point in building Gem of Isolation as she only has one ability that can reliably apply it, and that she already has a strong enough slow.
Rod of Asclepius is a very situational item that could be built if your team has plenty of sources of healing and you're the only magical god that can build it.
Polynomicon is commonly seen as a "core" item on Sol, and it really isn't. In previous seasons, it was more justifiable to build but this season, there are more and better options. This isn't to say that Polynomicon is bad but you shouldn't value it highly. This item has a place in a burst oriented Sol build, and should not be built very early (another common misconception), but instead, it should be built later in the build.
Book of the Dead can be a replacement for Bancroft's Talon if you value some survivability. Not an item I would recommend too much, but does have a place in Sol builds.
Pythagorean's Piece is seen as "the best mage item" by some however it is merely the one of the most cost efficient items, not the best item. This item can be good in Assault, or if you have room in your build for it and don't want any other item.
Hastened Fatalis is a strong item on Sol as it gives her great chasing, boxing, and sticking potential in addition to stats such as attack speed and penetration, but it isn't core. You can certainly play Sol without this item and isn't as core as inexperienced players will tell you. I will admit that Sol ADC will make better usage of this item that Sol Mid, because traditional builds of Sol Mid revolve around having high magical power to constantly burst your enemies, while Sol ADC wants to box. Hastened Fatalis should never be built as a 2nd completed item as it will harm your power curve, so it is much better to build it once you have a core power item such as Bancroft's Talon, but you can also build it later if you wish.
Spear of Desolation is another strong item. It offers a lot of great stats for the price. Ideally you'll build it for burst oriented or mid builds for Sol, typically being built as early as a 2nd tier 3 item and as late as a 4th tier 3 item.
Divine Ruin will usually not be built on Sol unless enemy healing is a massive problem, and your team alone doesn't have enough anti-healing or anti-healing application.
Spear of the Magus shouldn't be built on Sol because you'll only be able to reliable apply 2 stacks of this item's passive with Stellar Burst.
Chronos Pendant is not too great of an item for the reason that it's main appeal is CDR, which Sol does not need as much as other stats.
Demonic Grip isn't too great of an item this season seeing as how there are better options for Sol. Obsidian Shard serves Sol better than Demonic Grip, unless you have another mage on the team that can utilize the stacks of protection reduction you will apply.
Telkhines Ring is a strong item due to the stats it grants and the passive, granting Sol more DPS than most items would give, although the price makes it better as a late game option instead of an early option. Great item for Sol ADC and basic attack builds for her.
Shaman's Ring is a very underrated item. The stats it gives for the price are pretty nice, and the passive gives Sol nice sustain in fights and in lane. Personally, you can build it for Sol ADC as a 3rd or 4th item.
Work in progress...