According to Colliot (sorry Pete, I missed your warning that you had already asked him), Néron was a very nice person. The fact that he had so few students simply reflect the fact that at the time there was no pressure on doctoral candidate to produce something significant ever: the worse that could happen was that you would get $only$ tenure. Consequently, there was no need to find something significant publishable quickly and consequently doctoral candidates would meet their advisor about once a year to get some papers signed and that was all. In that situation, the choice of an advisor is pretty much arbitrary.

Colliot-Thélène chose him because he wanted to do geometry but not something too closely affiliated with Grothendieck. According to him, Néron had at least another student, but I have already forgotten his name. This other student computed Néron models of modular curves of genus 1.