First, read this, it has a lot of good stuff in it.  You never know what you might see in it that makes for a fairly easy fix for you.
Pat’s Big Ol' Mud Page (fixing muddy runs):
 
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
I sympathize with you.  Although my run is up on a small rise so it should drain and has a roof, it still turns into a muddy mess when we get rain for several days on end, practically all spring long.  It sounds like your coop is in a low spot where water runs to it, that has to be a mess.
I don’t think it’s in Pat’s article, but when I built my coop I added a few inches of clay dirt to the floor.  It sets on clay dirt anyway.  Once it sets a little clay works pretty well to keep water out, but if it gets wet and stirred up it can be a real mess.  I also put in a swale on the uphill side to divert rainwater away from it.  You might have to wait until the weather turns dry to do something like this, but it might be a reasonable long-term fix for you.  A lot of what you can do is going to depend a lot on your unique situation.  Things that work for me might not work for you.
You can try using straw, hay, wood chips, wood shavings, those palm fronds, or something similar.  Some people do and are successful.  But the risk is that they will get wet and not dry out so you might have to shovel them out and put some new in if the water just stands there.  
It is not healthy for chickens to always stand in wet.  It can damage their feet.  Just building it up with loose sand can help, but that’s often a temporary fix.  Over time the sand disappears down in the mud.  But a temporary fix may work long enough for you.  Putting sand bags on the coop floor may be a good plan.  Others use cinder blocks, plywood, pallets, something to get them a dry surface above the mud.  
I know this doesn’t do you a bit of good but maybe it will help someone else reading this that hasn’t built their coop yet.  It’s never a good idea to put a coop or run in low spots, but in some small back yards, some people just don’t have an option.  It can be tough.
Good luck!