For anyone interested, below is how I ended-up doing it without modifying SolrNet source code.
I've added an additional property in the class called IdAsString and set it as "BsonIgnore" so that it doesn't get stored in MongoDB, however I use that field to map it to Solr's id field:
[BsonId]
public ObjectId Id { get; set; }
[BsonIgnore]
[SolrUniqueKey("id")]
public string IdAsString
{
get { return Id.ToString(); }
set { Id = ObjectId.Parse(value); }
}