I realize that you are asking to increase Mog house storage and that is ONE of the "bags" that we have available for stashing items - but it makes me wonder if you were asking about Storage or one of the other bags. So, I will be my usual "answer more than what has been asked" and give you the links for expanding all the "bags".
Here is a link for all the storage areas that you access in the Mog House:
Storage would be the first bag, that is expanded by placing furniture in your Mog Safe. The Furniture you can craft, buy from an NPC or get from the Auction House. Pay attention to the number in the lower right hand side. That number tells you how many slots it will expand your Storage. There are furniture pieces, like the Panetiere, that have up to 65 storage spaces.
Mog Safe is expanded by doing a Mini quest involving 3 different beds that have to be placed in your Mog House and you interact with your Mog for this quest. Link below.
Mog Locker is only activated by doing a Staging Point and becoming a member of Salaheem's Sentinels, which is a quest that is part of Whitegate. Link below
Mog Satchel is received when you purchase a Security Token from SE to encrypt your account. I've been using one since the day they came out. I've never been hacked - that is the only recommendation I can give, I do not know anyone that has been hacked who has one. Do not ever let anyone have your token.
Mog Sack you get from the Artisan Mog for 9,980 gil, in the link below it will give you the locations of the Mog and further info.
Inventory or your Gobbie Bag - your main bag that you carry with you is increased with a series of quest out of Jeuno that don't come cheap, but is well with it in the long run. The size of your Gobbie Bag dictates the size of your Satchel. Any set of four items for one size expansion can be substituted with Goblin Stew 880, which may be available for 500 points in the Repeat Login Campaign.