As a GM, I love handing out props but I don't have a lot of time to build them. And as Melon points out (I've run some of his campaigns) portability and storage can be an issue. I've organically developed some question over the years, to help me analyse my props and make the most of the few times I use them.
Why am I adding this prop?
This is the big one. If I can't answer it, I forget the idea immediately (but see below for a variety of possible answers). A prop calls attention to itself, and thus calls attention to the story element it represents. It needs a reason to exist, or it's a red herring that will send the players running merrily after nothing.
Not that there's anything wrong with that, if that's the campaign style your group enjoys.
Will the group gather around it?
I love it when the players (not the characters, but the guys at the table) huddle around a prop to examine it. This usually means they'll pull it out again later, and cries of "Pass me the [prop], please," will resound for the rest of the session.
To this end, letters are my go-to props. They're easy to make (see below) but they also beg to be paid attention to, a series of letters even more so.
Does it further the story?
Very similar to my first question, but more specific. For example: Burying plot in the implications of the correspondence between two people, or even just using it to provide information about the relationship between two NPCs without having to talk to yourself (simulating their dialogue) is great, and rewards the players for careful thinking.
If a prop doesn't further the story, the next question becomes all-important.
Is it cool?
The Rule of Cool can forgive many sins of storytelling, and this is true of props as well. The cooler a prop is, the less any of the other questions here matter to me.
Can I make it (easily)?
Many GMs, myself included, don't have the time, resources, or skill to craft elaborate props from scratch. Found objects (things you pick up off the ground, get in junk stores, find moldering in a corner of your room) are often the way to go; I have an old-fashioned silver key that does regular duty as The Important Key You Found.
This is one reason I use letters so much: a fancy font, some judicious crumpling and ripping, and you've got a passable flavor item that the group will gather around to study. Writing them is the part you want to put your effort into, because that's the bit that is important to the story... though if you're clever you can use the wear and tear on the letter to say something also: blood? waterstains? ripped to pieces and stapled back together? These all say something about the content and the person who carried it. (Frankenletter was a big hit.)
What happens to it at the end of the session?
I have two categories of prop: single-session and adventure-long. A single-session prop might be a key that the party picks up, and then uses quickly. An adventure-long prop might be a series of correspondence that the party gathers as they explore, and which gets referred to over and over again as the story advances.
We have a PC Binder to keep our characters sheets in, and I design props to be able to fit in the binder's pockets so they're easy to keep track of over weeks and months. If a prop can't fit in the PC Binder, it's got to be a single-session thing.
How many props have I already introduced?
It's also important to think about how many props you're using. A deluge of props makes them all less important, and harder to track. I'd suggest an average of one prop per level is a prop-heavy game, assuming each prop is Significant and you expect the party to take time to examine it. If you're just using them as visuals and not expecting time to be spent in studying them, a prop each session could be fun.
Another reason I use letters is that they can tie into each other, making a series of five letters spread across five sessions feel more like a single prop being slowly unearthed.
Player reactions and bringing their own props
Some groups don't like props. They're fiddly, or they smack of LARP, or they take you out of the game immersion, or they slow down the party's door-kicking, whatever the party's pet peeves are. If the players don't bite, don't force the issue. If they huddle around the prop or pass it down the table, that's good. If they get it back out later, you're golden. Otherwise... maybe re-think the prop thing for this group in this campaign.
As for bringing their own props... this is tricky. Weapons are right out, even play ones, because it's really easy for someone to think a thing is safe when it isn't. I've had success with hats, though, especially in a game where players are running multiple characters: a hat for each PC helps everyone keep things straight. Hats stay on your head (and off the table), tend to be soft and harmless, and serve a purpose that advances the game experience.