Is it against the law to store a gun in a bank safety deposit box?
Yes it is. It is illegal to store a firearm or cash in a safe deposit box (I forget how many years exactly you can get in jail but it is quite a few). However, when I worked in banks and went to safe boxes with people it was strongly stressed that they were to be given privacy and we were not to see what was in their boxes. If you want to open in front of a teller or manager you can, and if they see the gun they HAVE to report you to the police. However, banks have stalls (sometimes rooms) for you to open your box in privacy. If you insist on privacy then no one but you will know what you keep in your box. Of course, I am only passing on information, I am by no means suggesting breaking any laws.
As for the bank box suggestion, law or no law, I wouldn't bother. You need to rent them on a yearly basis and some banks may require that you have an account (plus, these days some branches won't even have boxes). It has been a couple years since I've been a teller or account manager so I don't know today's prices, but 2-3 years ago a box large enough to hold a gun would go for about $50 around here.
For that you have a few other options:
-Rent a room in a storage facility. For one month you won't be over $50 and many have no minimum time periods and they have discounts on the first month so you
could be significantly under $50. Of course, it would be odd to have a full storage room for only one pistol.
-Call the hotel first and see if they have lockboxes for guests and see if they have a policy against using them for guns. Alternatively, if you want to be discrete about having a gun with you, check to see their privacy policies and then use a small unassuming bag to hold your gun if they don't check on the contents.
-Buy (or bring if you have one) a small quick access safe.
-Buy a small, cheap, Brinks safe or storage box. Mine, in addition to the lock, has a place to double lock it using a padlock. I could easily run a thin diameter cable through that spot or a bike lock through the pad lock and lock the box to something stable in the room.