As mentioned by Maître Capello, rendre visite is the set phrase. In using that phrase, you're referring to the concept of visiting someone. You're not so much referring to the visit itself.
If you add the indefinite article, then you are adding emphasis or bringing attention to the noun visite. Consider that if you were to qualify the word visite, say with an adjective or a relative clause, then you would put an indefinite article before visite. Or you would use another determinant in its place.
So if you hear the indefinite article, then your attention is drawn to the possibility that there is something particular about this visit that is worth expanding upon, like what snarkhunter mentioned above.