Weird things can definitely happen in real life. So how do you work these kinds of moments into your writing? It's a rather tricky concept, because it's easy to feel forced. There's a good chance the reader won't believe this sort of coincidence will happen in real life. And if they don't believe what's happening, then you've lost them.
First of all, you really need to assess whether or not a coincidence is even necessary in your story. If you can think of a better way to make something happen, then you should probably do it that way. A coincidence should really only be used if it is absolutely necessary, if there is no other way to bring your character to a certain place or realization, or even bringing two characters together. My second tip would be to have your character(s) acknowledge it. Have them think, "hey, this is weird," or even wonder if fate is somehow involved. It may seem cheesy, but if it's just a passing thought and you don't hammer it over your reader's head, then by all means, play the fate card. Or heck, call it what it is--a coincidence!
I think my biggest bit of advice would be to not overdo it. Unless your entire book is about coincidences or the power of fate, you should probably only have one or two coincidences in there (if you need them at all). If you make something unbelievable happen over and over again, the reader is going to get annoyed. They're going to find themselves saying, "This would never happen!" And again, unless that's the point you're trying to make, this is not good.
Example time! I think (or hope, anyway) that I only have one coincidence in my WIP. Basically I had to get one character to another character's apartment for a pretty important scene, but with one major problem: he didn't know the address. In the short story version, the other character gives him his address, but as I reworked things, I realized this would never ever happen. So I had to think it over. I didn't really think this scene could happen anywhere else, and having the element of surprise when he just shows up at the apartment seemed to help. But where does he get that address?! I realized there was another character who had it--the MC's mother. Of course, he couldn't just ask for it. I thought about him rummaging through drawers and address books trying to find it, but that seemed silly.
Anyway, I'm rambling as usual, so I'll cut to the chase. Here's how I did it. Throughout this scene, my MC, Jordan, has been talking with his mother and the whole time she's been putting stamps on a stack of envelopes. He gets annoyed with her and announces that he's leaving. Then this happens:
She threw the stack of envelopes at me. “Drop off the mail, then.”
I would have glared at her but she wasn’t even looking, so I reached for the envelopes with one foot pointed toward the door. And then I had one of those perfect moments. If I hadn’t looked down for just a second, it might not have happened. But clearly it was supposed to happen, since the universe so conveniently placed the answer right in the palm of my hand. I mean, I’m used to getting my way, but this was more than that. This was a sign. Because guess whose name was on the top of that pile? And not just a name, but an address.
Now, I certainly don't claim to be perfect with my writing, but I think I worked this coincidence in the right way. He acknowledges how this moment is weird and convenient, a sign, even. But he doesn't over think it. Once this paragraph is done, he doesn't mention the coincidence again. He just acts. He goes to that address, and the story moves forward.
That's what a coincidence should do, really. It moves the story forward when it reaches a road block. You don't want to spend pages and pages over-analyzing the significance of this coincidence. Just let it happen, acknowledge it briefly, and move on. Let the coincidence do what it needs to do and then forget about it. Sure, sometimes coincidences are unbelievable, but you still need your reader to believe it.
How do you guys deal with coincidences? Do you try to avoid them? Any examples in your books or ones you've read you'd like to share?