I think MongoDB has a lot to offer DH projects -- it basically incorporates some of the best features of a document-oriented database and a relational database all in one. For so much of the data often worked with in DH projects, that are sometimes squished and squeezed so much to fit a relational model that never quite works, a schemaless approach to indexing and storing data like Mongo has works very well. I especially see value for those projects that don't need a full-blown structured repository like FEDORA, but that need more than flat files or relational DBs can do.
Having said that, the biggest thing to consider is that you say using MySQL works very well already for you. If you can't tell that there is something missing, not-quite-right, or not efficient enough with your current setup, the transition to Mongo may not be worth the effort in the long run. Mongo is pretty straightforward to set up, but it does approach data differently and doesn't use SQL at all, so when you say you'd be rewriting the backend, you'd really be rewriting it! You have to evaluate your data itself to know if the effort is worth it. Is your data tabular, or document-based? Are you mostly dealing with numbers and simple structured data (with some relationships), or heavily semantic data with complex metadata?
Perhaps another path to consider, if you decide to rewrite your back end, is to completely decouple it from your front end. Create a thin layer that is nothing but a RESTful API -- the api talks to the DB and only spits out structured JSON on the other end. This would also likely require some tweaking of your front end, but it will A) make it so you can then very easily write multiple other front ends (mobile, TV, etc) without a hitch, and B) 3 years from now, if you decide to change out the DB layer again, it's much easier to then create a new API generator that will offer the same data structures than it is to create an all new backend interface. There are lots of starting point solutions for quickly generating RESTful APIs whether your DB is MySQL, Mongo, or something else.