Database
[{words:{wordOne:["Cat","Dog","Fish"],wordTwo:["Red","Blue","Green"],wordThree:["Square","Cirecle","Triangle"]}}]
Query
db.collection.aggregate([{$project:{all:{$reduce:{input:{$objectToArray:"$words"},initialValue:[],in:{$concatArrays:["$$value","$$this.v"]}}}}},{$unwind:"$all"},{$project:{_id:0,sampleWord:"$all"}},{$sample:{size:1}}])