I'm not entirely sure if it's an actual abbreviation thereof, but I believe なんての as a set casual expression is equivalent to などというの, and essentially means "something like", "the likes of", etc.
相談 and 事 aren't really separate words here; 相談事{そうだんごと} is something of a set expression that means roughly the same as 相談 on its own; I'd say the difference is just that 相談事 sounds a little more concrete, like it's about a specific topic or a more formalised discussion.
個別 doesn't really mean "different" or "special" here but rather "individual" - he's specifying 個別の相談事 "an individual, one-on-one discussion" as opposed to a discussion involving everyone.
And finally the のも - this is linking back to the のも in the previous sentence. This whole final clause is a clarification of what he means by こういうの in the previous sentence, so it's possible to rephrase them as a single sentence へー、個別の相談事なんてのもアリなんだ "Wow, so individual consultations like this are allowed too, are they?"
Separately, the two sentences could be translated as "Wow, so stuff like this is allowed too, is it? You know, things like individual consultations." (The explanatory さ particle in this instance fulfils a similar role to the "you know" in the English sentence, helping to mark the final clause as a clarification of the preceding one.)
