Even though the Player's Handbook says a dragonborn only gets one breath weapon per short rest, is there any way outside of rule 0 for a dragonborn to gain multiple uses per short rest, like by using a spell or magic item?
"It states in the book that the PC would only get 1" - I think there's your answer. Should you wish to wander into the realms of homebrew I'm sure a DM could find a way of allowing a dragonborn to gain an extra use of breathe weapon - perhaps instead of a skill increase at level 8.
If you want to stay more true to RAW then I believe potions of fire breath make a good breath weapon substitute.
Yes. There are 5 magic items in Tyranny of Dragons that if the user has a breath weapon that recharges on a rest it has recharge of 6.
A dragonborn can only use its breath weapon once per short or long rest. This is very important, because like a Warlock with their spell slots, in order to stay relevant throughout most of the adventuring day, you need some short rests (same with the Druid and their Wildshape ability).
That being said, DM's aren't required to give you short rests, although I believe it is stated somewhere in either the Player's Handbook or the Dungeon Master's Guide that the classes are designed around two short rests a day. This question has some good answers in regards to how often short rests should happen and more, I suggest giving it a read.
So, having more short rests will allow you to use your regular breath weapon more often. And, potions of fire breath will allow you a temporary additional breath weapon (I believe it's three uses). However, if you want a permanent extra breath weapon, there is the Variant:Mixing Potions rule in the Dungeon Master's Guide on page 140. If your DM uses this rule, it's possible (though highly unlikely) that drinking a Potion of Fire Breath and another potion together could result in one of them becoming permanent, although the exact mechanics of such a thing would be left entirely up to your DM. Me personally, I would likely make the permanent effect of the Fire Breath act similar to the Dragonborn breath weapon, not only for simplicity, but also so that the natural breath weapon of the Dragonborn stays just as relevant. This would essentially give you two breath weapons per short/long rest, which I thing would work just fine.
And of course, Homebrew is always an option, such as having a feat that allows you to use your breath weapon more than once, or spells or magic items with similar functions, or even homebrewed potions that work in the same vein as the Potion of Fire Breath (which you could use with the Variant:Mixing Potions rule) but with different damage types.
A few months after this question was asked, Xanathar's Guide to Everything was released. It includes a new spell called Dragon's Breath. Dragons Breath is a level 2 spell with a duration of 1 minute (concentration). It takes a bonus action to cast, and it allows the target to do 3d6 damage in a 15-foot cone as an action.
This spell gives you a breath attack that covers the same area and damage as a level 6-10 Dragonborn (Gold, Green, Red, Silver, or White). Because the spell is a bonus action, you can cast the spell on yourself and then immediately use the breath attack. It's much better for regular use than a potion of fire breath, because level 2 spell slots are much easier to come by than expensive magical potions.