Cost-effective items for mana-hungry AP champions
Solution 1:
This is a very situational, and depends on your play style.
Most skilled AP Mids don't get an item based on whether it gives you more mana or mana regen. Athene's Unholy Grail (AUG) is such a good item for mana hungry AP champs such as: Ahri, Anivia, Kassadin etc. not only because AUG has amazing mana regen abilities, allowing mana hungry champs to have sustain, but it also gives Magic Resist (MR) and a decent Ability Power (AP) boost. Everything an AP mid would need to succesfully win his lane. It's also a very easy item to build up too.
Rod of Ages (ROA) is a good item because it gives mana and health and ability power. If you build this item first, Catalyst the Protector gives a decent sustain with its passive ability. Also being one of the few meta acceptable items for an AP mid to build that has health, its great for survivability
Now a days the only Champion that usually builds Archangel Staff is Karthus, and its usually 4 Archangel Staffs with Sorcerer's Shoes and Rabadon's Deathcap. This usually accomplishes the goal of being able to nearly one shot the entire enemy team with Requiem. Although I wouldn't suggest this build at all.
As a rule of thumb for my builds I usually go heavy mana regen for AP Champs such as Ahri and Kassadin. Champs that have long cooldowns, and burn through their rotation very quickly. Heavy Max Mana is great for champions like Anivia and Karthus who have very important AOE Spells that are channeled and tear through mana faster then any amount of Mana regen will keep up with.
But this is hugely dependent on play style and preference and my opinion.
If you'd like examples for more champs let me know.
Solution 2:
I think Chalice of Harmony is a very good item for AP mid combined with blue buff this will give you almost unlimited manna, and you will also get a bit of MR which is very handy.
This item also can be upgraded and is very handy late game.
Solution 3:
My personal recommendation:
Rod of Ages: I like this on almost any AP champ that needs mana because you get a good amount of AP and good chunk of health which you will definately need late game if you neither want to go completely glass cannon or sacrifice an item slot for a defensive item (veil etc). Get this relatively early to max it out by mid game and it's a safe choice for most AP casters.
Notes:
Athene's Holy Grail is also a good choice if you also need the CDR and are willing to sacrifice some AP AND there is at least 1 good (or fed) AP caster on the other team. But I don't usually go for it. The reason is you can get CDR from other items that give you greater dmg output plus we all know that the health from ROA is more efficient defensively than the +40 magic resist.
Archangel Staff might be a choice but it requires commitment to building more mana items which could lead to a weird build. I definitely don't recommend building 4 of them. The only situation where I find it useful is if you wanna build as a tanky AP caster. There is some potential there because you can build (some of) the following items: Archangel Staff, Rod Of Ages, Frozen Heart, Banshee's Veil, Wooglet's Witchcap and Boots of your Choice. This build would give you tons of mana to spam your spells (with good cdr from Frozen Heart and possibly boots of lucidity), decent AP as well (mainly because of the mana combo with Archangel Staff) and insane tankiness both health-wise and armor-magicresist-wise.
Solution 4:
Your question title doesn't exactly match the body of your question, so I decided to help answer the question in the title.
I made a spreadsheet with every item that gives Mana or Mana Regeneration on it. It lists MP, MP/5, gold cost, as well as some calculated values that should help with determining what to buy. The attributes given by certain items will depend on various factors; thus, the spreadsheet has a Constants sheet that you can edit to see various things.
Obviously, there are many factors to consider when buying an item; this will only help determine which Mana or Mana/5 items are more gold-efficient.
Note: I added consumables for completeness' sake, but I didn't add Ichor of Illumination since I couldn't find any data that quantifies "a huge boost to mana and energy regeneration."