Just as a disclaimer, your session style might require different props than I discuss here, and that is ok! Ultimately you and your photographer can do whatever you want!
Phew, ok. With that out of the way, let's talk about props. There are good props, there are bad props, and their are cliche props for a typical photography session, and by that I mean family photos, engagments, senior pictures, bridals, etc. Any other specialized shoot falls under a different category and this post will not apply as much to those shoots!
Good Props
Over the years as I have researched different photography styles and looked at many different collections of photos, I have come to the conclusion that memorabilia in a photo is super neat. Memorabilia can include items that represent family members, such as flowers or their picture; it can be something special to you as an individual; etc. Hopefully that wasn't too broad of a definition for you to get the idea.
One of my favorite sessions that had some good props was of a family whose 2 oldest sons were out of the country on missions for the Church of Jesus Christ of Latter-Day Saints. They had printed some giant cardboard cutouts of their heads to include in some of their family pictures! So fun!
Other good props can be basic pieces of furniture to use, such as bar stools or chairs or blankets. Maybe find a different stool to use than a piece of cinder block, though...
A popular way to do senior pictures is include props that show about what you are interested in, such as your instrument, your Letterman jacket, If you did a sport in high school you can bring a memento of that (trophy/medals, shoes, helmets, equipment, etc), or wear part or all of your graduation cap and gown.
Another type of session that I have done a couple times is for missions or baptisms. Some props that are great for these are scriptures, CTR rings, etc.
Cliche Props
For a typical photo session that is not stylized or for a magazine or anything, there are some props that are just unnecessary and a little bit cliche. Avoid the unnecessary if possible, but include it if that is the look you are going for! I will not say any items by name, because you and your photographer can decide what should be included for the style of photo.
Props are tricky! That is the moral of this story. But, just because they can be hard to figure out does not mean they should be left out all together. These are your photos, don't forget that.
Best of luck to you! Have fun, be brave, create memories.