Generally, using a prepackaged library will offer benefits early on in a development cycle where you can make rapid progress by not concentrating on the details and muck that can make building a proof of concept application challenging and time consuming.
If you're just building a PoC, then often it doesn't matter what the specifics are of the construction, unless you also need to validate architectural choices. For example, it may be that it's easy to wire up a simple document model with your iOS application, but adding a layer of security on the existing framework may be difficult (I'm saying in general). Or, that by not giving enough attention to the data model, that it later becomes difficult to model the database in a relational database, or that even when staying with the same technology that was used in the PoC, that it won't scale well under load).
By building or extending your own Restful API, you'll have complete control over the entire experience. Many web platforms today make building a restful API quite natural so that the developer can concentrate on the application logic, rather than the plumbing of the architecture. So, you may not need to take on a complete end-to-end framework solution.
Frameworks are often opinionated, often to not any official specification. Once you've moved your code beyond a PoC, you'll want to decide where the risks to a production system are. How many moving parts and widgets are there? What if something goes wrong? Will you be able to understand the interactions? Is support available for the platform? Are the libraries actively maintained and are the issues on their support list long, solvable, etc.?
The challenge of using a document oriented database like MongoDb or CouchDb will be technology transfer if you decide on an alternate database platform later. If you need to adjust, consider how intertwined your business logic and database code is with the various frameworks you've used. It's the same problem if you start RDBMS, and try switching to a Document database. Converting a data model between the two can be very complex regardless of the starting point.
For a PoC you're going to throw away, use what gets things done quickly. If it's more, then you'll need to consider how it fits.
Eve has a small issues list, is updated frequently, and has good documentation. The ultimate decision is up to you.