"Se" refers to the verb, it is part of it. You need it in order to express the meaning of the verb, so you use it in any of the persons:
–se me ocurre
–se te ocurre
–se le ocurre
–se nos ocurre
–se os ocurre
–se les ocurre
As you see, "se" and "ocurre" does not change and the indirect object is what actually changes depending on the person who talks, the grammatic subject of the sentence. Remember! the gramatic subject in this kind of sentences is not yo, tú, él, ella, nosotros, ellos....It is A mí, A ti, A él, A ella, A nosotros, A ellos:
–A mí se me ocurre
–A ti se te ocurre
–A ella / A él / A usted se le ocurre
–A nosotros se nos ocurre
–A vosotros se os ocurre
–A ellos / A ustedes se les ocurre
The actually subject of the sentence it is always something, a third person thing, that´s why the verb it´s always in third person:
–A mí se me ocurre una idea ( Una idea, is the Subject)