ronrsr, how true. And if you haven’t read the etiquette book the simple thing is just to say “I am so sorry for your loss” and leave it at that. Or in the case of an illness “I wish you (or your family member) a refuah shlemah b’karov”. You can’t go wrong with that either. Of course if you need to take it further, you can always ask for the name so you can say a kapital Tehillim, and then you can leave it at that.
The other appropriate thing to do, and only if YOU REALLY MEAN IT, is to ask if there is any that YOU can do to help or assist. If you do mean it, write your name and number down on a piece of paper and give it to the person saying “I know a lot of people offer but I really mean it, if you need me you can call me.”