Aboru aboye aboshishe,
Clearly this is a difficult question. Let me first note that there is a difference in how initiations work (in Trad Yoruba, you receive you guardian or ruling Orisa and Esu only, in lucumi, you receive 6 or more all at once, simply an adaptation for survival that occured after the middle passage). This has been a long running issue in the Orisha community in the diaspora. In the absence of a patrilineal or familial link, how can one determine which Orisha rules their Ori?
The answer is, as often is the case in traditional african practice, that there is more then one way. In general, it is accepted that when a child does their Esentaye through Ifa, this will be determined and the child will know their path in life. In the absence of Esentaye, as many either come to the practice later in life, are not Yoruba, or do not come from a family which practices Orisha worship, one can consult a Babalawo or as called in the Lucumi system have a "Plante" with 3 Babalawos.
The Logic here is that as Eleri Ipin (witness to creation), Orunmila is present as our Ori chooses it's path/destiny at the foot of Olodumare. As the only Orisha with direct access to Odu (in which is housed all knowledge of the world past, present and future), Orunmila can fully determine what the path an individual chose, as well as what important steps they need to take to achieve that path (which might include Adosu / Kariosha). While Orunmila does not have the direct power to control Lightening, like Shango, or the Wind like Oya, or the Ocean like Yemoja, Orunmila can consult Ifa/Odu to determine what someone must do in order to affect their future. That is Ifa/Orunmila's role.
We can see that Orunmila throughout Ese, Oriki, Iyere, Pataki, acts as the neutral arbiter in many of the arguments that Orisha have amongst themselves. He also acts as their counselor and as such displays his unique role in maintaining the balance in nature and life.
That all said, after conversations with Babalawos and Olorishas in Nigeria, there are other ways in the Traditional practice in which an Orisha can claim someone's head. The most significant of course is possession. When a lay person becomes mounted by an Orisha, there is no clearer sign that the person is not only claimed by that Orisha, but will need to be initiated to the secrets and priesthood of that Orisha. This is also true of the lucumi.
Lastly, it should be noted that while Ifa is considered to be the optimal way to determine a person's ruling Orisha, this can also be done in Traditional and lucumi practice by a competent priest/diviner (Olorisha) of the cowrie shell method. As a rule of thumb though, the first two methods described here are considered to be the optimal ones in determining a person's ruling Orisha.
This of course has ignited a bit of controversy in the diaspora as Olorishas and Babalawos tend to clash and view the process more as a power struggle over who's right, and who has the right, then what's best for the devotee. As a Babalawo, while I am of the opinion that all three methods are legitimate, the first two are the methods I would consider most reliable. As argued earlier, I come to this conclusion because:
a. Orunmila is the diviner who's sole job it is to interpret the Oracle of Ifa and help guide people on their paths to attaining their destiny. Let us be clear, Orunmila is NOT all powerful, and does not himself determine peoples destinies, he can only tell people what is in store, and determine Ebos that a person can do. He does have the secrets and knowledge to get Esu, the Orisha, Iyaami, etc to help him with things, but he himself does not do these things directly.
b. Ifa/Odu contains all the knowledge of the world, past present and future, and as such supercedes other methods of divination. This isn't arrogance, it's simply an acknowledgement that by definition, if you accept Ifa/Orunmila, you accept it is the highest method of divination.
c. Orunmila is the arbiter. He often is consulted to maintain order amongst the Irunmole/Orisha and as such can play a neutral role in decisions.
d. Trance possesion or "mounting" is when the Orisha has decided to manifest directly on earth in the body of a person. That choice by the Orisha means that they have chosen you as their medium and that you belong to their priesthood.
One last note. Unlike the diasporal lucumi practices, in traditional African practice, being mounted by one's Orisha does not preclude them from the Ifa priesthood. I've had extensive discussions with my Oluwo in Nigeria and others, and if Ifa calls you to his priesthood, you must go. There exists in the Lucumi system ceremonies that can be done for someone who has mounted Egun to pass to Ifa, and ensure they aren't mounted again. Once initiated into Ifa, through the rites/rituals that occur, one will no longer be mounted by their Orisha afterwards. I once heard it described as Babalawos are in a constant state of possesion by Orunmila/Ifa which effectively keeps any other possession from occuring.
Also, of interesting note, while in the diaspora, being initiated to one's teutelary Orisha after passing to Ifa can not be done, that is not the case in Nigeria, and one can be initiated to their guardian Orisha after passing through Itefa (initiation as a Babalawo).