Warning Spoilers!

The Children

I recommend you read the Evangelion link first before reading on.

Theory

The children are actually numbered according to their level. Level you may ask? Well for instance it was believed that the numbers (i.e. First Child or Children) were according to who came first. Actually it is their level of synchronization per say. Ok...Ritsuko says that Touji is the fourth LEVEL child. Asuka asks why him? And Ritsuko answers: All the children in Shinji's class are fourth class children. They are all eligible to be pilots. Why Shinji's class? I guess his grade level...remember the whole 14-15 year old thing? That's probably linked to it.
Thus:

Rei is 1st because she is an angel.
Asuka is 2nd because her link to her mom and her training.
Shinji is 3rd because his link to his mom and no training.
Touji is 4th because he's the best eligible out of Shinji's class.

Now about the kids being 14-15....I really don't know my best guess is that has to do with being born of the second impact and Shinji's class eligibility thing......I dunno.

UPDATE The pilots of the Eva's all seem to be 14-15 year olds. They were all children of the 2nd Impact. 2nd Impact was caused by Man's first contact with the Angel. In order to decrease any further damage caused by the unfolding impact they shrank Adam. After the initial contact, human beings were now somehow linked to a chain of events. Possibly the events foretold by the Dead Sea Scrolls. The children born of that time were somehow linked to the Angels or just to Adam itself. Eva is a clone of Adam. Thus the children have a link with the eva's and are able to synchronize.

Reader Theories:

The reason that the children can pilot Evangelions could be hereditary. Shinji is able to pilot EVA-01 because his mother was a pilot of the EVA. Asuka can pilot EVA-02 for the same reason. Rei can pilot EVA-00 since Yui Ikari piloted an EVA, and Rei is a clone of Yui. It may also be because of her Angel DNA. Kaworu can pilot EVA-02 because he is an angel. I guess two simillar organisms (EVA and Angels) are compatible somehow. This does not explain Touji, however.
-Patrick George

The actual reasoning behind the selection of EVA pilots is that they are all motherless children. In episode 4, Kensuke mentions that he has no mother, as Toji also indirectly mentions in the previous episode. Also, Hikari and her two sisters are the only ones left and, thus, have to fend for themselves. The evidence to support my theory is there, you simply must pay a good deal of attention.
-Sebastian Morris-White