I don't recommend picking a Doran's Ring for an AP mid start. I explain more why later.
However, stacking Doran's Rings has a different purpose. As was said before Doran's provides you overall with more stats for gold than other other items for the same price. For example, Doran's Ring gives a bonus 80 hp and some mana regen for only 5 AP less than an Amplifying Tome. So for champions that have high AP ratios (so that 15 AP goes further), need some hp, and are heavy on their spells the extra rings can help you get ahead in the short run and then snowball to victory.
For example, champions like Amumu use Doran's Rings very well. Amumu chews through mana pretty quickly, and his ratios of 1.0, .01 (does % life damage), 0.5, and 1.0 use any bonus AP super well. It allows Amumu to rack up some damage and then focus on other items that he needs like tanking, because he does very well with little AP and some bonus mana regen. It's also important that every item on Amumu provides some bonus hp as he has to be in close range and take some damage in order to cast his spells.
A champion you wouldn't want to use it on is Mordekaiser. The benefits from AP, but mana regen is completely useless on him. Also, bonus hp is less useful than bonus armor and magic resist, because Mordekaiser's passive shield becomes more effective with armor and magic resist. So anything you can do to increase the power of his shield will negate the need for little hp boosts on every item Mordekaiser uses.
So for pickign a starting item, compare what Doran's ring gives you for a start versus Boots and 3 Health Potions (both costing about the same and possible to buy without masteries):
- 80 effective hp
- 15 AP
- 5 Mana Regen/5
Boots and 3 Health Potions:
- 450 effective hp
- ~60 bonus movement speed
To explain effective health quickly, basically Healing Potions do not give you actual hp, but they heal quickyl enough that you effectively do have additional hp. The exception is when you receive lots of damage in a short time with healing reduction the Potions will not save you. However, that is rarely the case in early laning, and even with healing reduciton Potions heal 75 hp (almost the same as Doran's provides) in about 6 seconds. Unless you make a very silly mistake, Healing Potions do really provide an effective 450 hp.
That 80 hp only soaks one additional spell at level 1, and even less at higher levels. Compare that to 450 hp, which soaks 4 spells at level 1, or about 3 at slightly higher levels. Already the boots provide a pretty heavy hp advantage over the 80 hp from Doran's because the Potions heal a large amount of health. Also, keep in mind that with Boots you will be able to dodge skill shots easier (which compose a good portion of some AP mid's spells) and you will be able to stay out of their range, preventing them from casting spells at you or harassing you with auto-attacks. Champions like Annie will chew through you with auto-attacks at early levels if you cannot keep out of her range; she has an incredible 650 attack range that few other auto-attacks and spells in the game can out gun.
So really what you get from the Doran's is the 15 AP and mana regen. First off, most AP mids only have AP ratios of 0.9, meaning that's an additional 13.5 damage per spell. That is pitiful at best because it will take 6 spells for that bonus damage to match the damage of any single level 1 spell, let alone a level 3 spell. ALso, it doesn't add any power to your auto-attack, which can be a good portion of your damage. Second, the mana regen is basically useless - either tool some runes to help you with mana regen or adjust your playstyle so you consume less mana. Learn to last hit with auto attacks instead of spells, and save mana for kills.