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.