Q:

What is the cost/benefit of using mongoid versus activerecord?

Google searching leads me to questions looking to use mongoid and activerecord in parallel. I've seen a lot of questions specific to mongoid and a lot of questions specific to activerecord, but none comparing them.
I guess the question is, what are the benefits of running the mongoid adapter with rails for mongoid db versus running activerecord with mysql?

A:

Mongoid and ActiveRecord are both database ORM (technically speaking, Mongoid is a ODM, but that's another topic). However, Mongoid targets only the MongoDB database, while ActiveRecord targets relational databases such as PostgreSQL, MySQL and SQLServer.
For this reason, it's not really possible to compare them. In fact, they are not one a replacement of the other. Which one you use, depends on the database behind your application.
You can compare ActiveRecord with DataMapper or Sequel. Likewise you can compare Mongoid with MongoMapper (actually you could, but there is no comparison because the last one is almost dead).
For this reason, I believe the question doesn't have sense in that way. I assume people would choose the ORM because of the database, not the database because of the ORM.

A:

You are talking about Object Document Mappers (ODM) and Object Relational Mappers (ORM).
Mongoid is an ODM for MongoDB.
That is a NoSQL database.
ActiveRecord is an ORM for PostgreSQL (and a few others).
This are SQL databases.
So the real question for you is:

Should I use a SQL or a NoSQL database?

Once you answer that, you can pick an ODM or ORM.

