It is a bit of an idiom, meaning "of the people". I think it means that he is socialite i.e. a member of high society, always going to parties, must be seen in the right places with the right people. (Does this fit with the character in the book?) This is not a common thing to say today, as (to me) it sounds quite posh, upper class, and a little old fashioned. I can only imagine this being said by someone in the British upper classes! There is a similar expression: "he is a people person", or (rare to hear in practice) in plural "they are people people" (i.e. no " 's " on the first "people"). This has a different meaning. It means someone who likes being with people, is good at dealing with people. For example, to work in customer service or as a nurse, you should be a "people person". This is quite common to say, and is a positive expression. (You might even write this on a job application.) However, I don't think that this is the same meaning used here in Dorian Gray.

Nuisance is a word that describes someone who is annoying or tiresome. I believe, although I cannot say for sure, that a people's person is someone who's entire appearance is built around pleasing other people. In this sentence, Lord Henry is making a generalization and saying that people's people in general are nuisances. He is saying this in reaction to someone else telling him that this specific person may not be able to come. Lord Henry is annoyed that this person can't come, and he expresses his annoyance by saying "what a nuisance people's people are!" meaning that people's people are annoying. I hope this helped!

I think here that "people's people" are those around one that make demands of their time (e.g. family, friends, etc.) Lord Henry would like his invitation to be prioritized over those of others. my take anyway.