I am having trouble finding a NoSQL databases that officially support MonoTouch via a local DB on the device. If their are, could someone provide a list of them here.
According to http://nosql-database.org/ there's siaqodb. Note that others might support MonoTouch without being mentioned in that site.
Edit: a few more clicks shows that HSS Database (from the same list) also supports MonoTouch.
You might also want to look at which ones support iOS (e.g. with Objective C) and see if bindings are available (or write your own).
A bit late, but still relevant:
I'm the author of MarcelloDB, and I just released version 0.3.0 on nuget.
MarcelloDB is a document DB, built specifically for mobile apps (light-weight, low memory usage) and supports Xamarin Android and iOS as well as the windows platform.
I still have some features I want to add before reaching v1, but the file format and existing api are allready quite stable.
RavenDB supports an embedded mode, and can run on Mono using the "Munin" storage engine option.
Although, there has been talk in the user group lately about dropping Munin, and it's not usually recommended for production, so it may not be a viable option.
I've not heard of someone using it with MonoTouch specifically, but there are some running it on Mono. If you try it, please update comments here with your findings. Thanks.
Take a look on Couchbase Lite xamarin's component