Why wont Cid take my items to be upgraded?
Cid won't take items to upgrade - He just makes remarks and complains that his back hurts.
What do you do to get him to take items?
If Cid isn't taking your weapons, make sure that he doesn't already have a full que of weapons he's working on and that you've waited the proper amount of time for those weapons to be upgraded.
Another reason Cid may not take your weapons to upgrade is that you have no weapons he can upgrade as there are only specific weapons that he can tinker with.
Cid can only upgrade 2 weapons at a time initially. After this I believe he only upgrades 1 weapon at a time.
To speed things up you can complete some relatively easy quests, or the nearby hunts and then sleep at the trailer. You should get a call off Cid telling you your weapon is ready for collection and you can set him off upgrading your next weapon.
It's worth noting that you should ensure you have the items he needs to upgrade your weapons. You can see what these are by hovering over the quest in your log and reading the summary.