How do long string field names affect MongoDB database size?
This is covered in the developer FAQ, some relevant excerpts:


  MongoDB stores all field names in every document. For most documents,
  this represents a small fraction of the space used by a document;
  however, for small documents the field names may represent a
  proportionally large amount of space


And, just a note on indexes:


  Shorter field names do not reduce the size of indexes, because indexes
  have a predefined structure


So, yes, reducing your field name size will make storage more efficient, though it will have no impact on index sizes.  Whether the saving you will make is worth it (versus loss of descriptiveness) will be up to you.  As an approximation, you will probably save something like 16 bytes per field if you drop to 2 character field names from 20 (for example) and that should mean that the document size will be reduced by more than 40% (~400k).  

Here's an easy way to estimate this using the MongoDB shell:

$ ./mongo --nodb
MongoDB shell version: 3.0.2
&gt; testObject = {"12345678901234567890" : 1.23324}
{ "12345678901234567890" : 1.23324 }
&gt; Object.bsonsize(testObject)
35
&gt; testObject = {"1" : 1.23324}
{ "1" : 1.23324 }
&gt; Object.bsonsize(testObject)
16
&gt; testObject = {"12" : 1.23324}
{ "12" : 1.23324 }
&gt; Object.bsonsize(testObject)
17


The Object.bsonsize method will give you an approximate size of any document in bytes but does not include padding, indexes etc. that would actually be used when storing a document in the database. Hence, these are all very approximate numbers - I would recommend testing with actual data to get a more definitive example.