their

The correct answer is "his/her" or "his or her". However, most native English speakers will say "their", as in "A person who is only interested in their own troubles is not your friend". This is not technically correct but is generally accepted and used.

A person who is interested in your own trouble is not your friend

You would say "his or her". In writing, you could write "his/her". For example: A person who is only interested in his/her own troubles is not your friend.

This is actually a relatively controversial topic in English grammar. Traditionally "his" was used by default: A person who is interested in HIS own troubles is not your friend. However, modern usage (which is sensitive to gender equality concerns) tends to favor either "his/her" or "his or her" or, quite commonly "their" (A person who is interested in their own trouble is not your friend.). Some language prescriptivists (those opposed to change/evolution in language) feel the use of "their" in this case is incorrect. But apparently using "their" in this manner goes back at least to Shakespeare's time. My own view is "their" is slowly but surely becoming dominant in this case - - at least in N. America. Great question, by the way!