It’s also a place to keep a deposit safe. “Safe” can be used as a modifier (e.g., “safe haven”). So you start with a box. A box where you put in money is a “deposit box.” One where your money is safe is a “safe deposit box.”
“Safety deposit box” comes from the mistaken beliefs that “safe” is only a noun, and that a noun cannot modify another noun.
In any case, both terms are used interchangeably, regardless of which is “right” or not.
Looking at updated cites in newspapers and books, the term were interchangeable around 1870. Both meant the same thing.
Certified Master Locksmith here.
It’s safe, not safety. 99% of the locksmith supply companies, locksmith magazines, and locksmith catalogs all call it a “safe deposit box”. It’s a box which is inside a safe, where you deposit your valuables. There may be confusion amongst the general public, but professional locksmiths are in near-unanimous agreement on this issue, to the point where you could chalk up the rare occurrences of “safety” as mere typos.
The Wikipedia entry is listed as “Safe Deposit Box,” but it says “otherwise known as safety deposit box.” American Heritage Dictionary has an entry for safe-deposit box but does not give an alternative name.