In the situation you describe, I would try looking at the real-world occult systems instead of fictional magic, since they most often try to explain, why anybody has a potential to do magic, but only a small number of their adherents can be successful. Folk tales are also a good source.
Magic is taught
I will mention it only for the sake of completeness - you've already stated that people can discover magic by themselves. There basic idea here is that magic is either very complicated ritually, or magical thinking is so significantly different from everyday thinking that you need to be taught to use it.
Magic is done by spirits
It is pretty popular too, and it may be not the best for your setting. In this version, a source of magic is not exactly in the person, but in his ability to call upon spiritual entities and coerce them. Maybe there is a shaman-style spiritual quest, whereby the prospective magic users discovers his spirit helper and then it teaches him further. It may be a matter of knowing the Names of the spirits (<PERSON> style), and knowing at least one name puts you into position to discover more.
Magic is gifted
This version also presumes the existence of the spiritual entity . There simplest example is the folktale idea of the 'deal with the Devil', 'deal at the crossroads' etc. Magical ability is conferred by a spirit as a part of some bargain. 805
It doesn't need to be a devil, it may be an ancestor spirit looking for vengeance.
Magic is insane
That is my favorite. It is used in fiction too, but mostly in the 'magical realism' style of writing, not in the mainstream fantasy. In this version, magic exists in the world, but it's illogical and irrational enough so that an average person is just not able to think this way. Something similar to the theories of quantum physics, for example. 'The world is not what it seems' - the human understanding of events and things is locked into linear time, the illusion of actions and consequences, or the illusion of free will and similar stuff. A practitioner needs to pierce this veil of illusion in order to grasp the real laws according to which the world operates. It is a bit like Zen Buddhism this way. Such enlightenment may be a result of a long study under another practitioner, but may also just happen on its own - either as a result of meditation, or a particularly traumatic event. In such version, all magical practitioners would seem slightly mad and unhinged to the ordinary people. 805