You say you want to put up a stone on his grave. Is it a public or church cemetery? They keep records of who is in the graveyard and when they were buried. Since he died in the 1940's, over 50 years ago, most states allow you to get the death certificates for genealogy reasons. I'm assuming he is a relative if you want to put a stone up for him. If he owned property, will or not, there should be probate records at the courthouse that will tell when he died. They will also most likely tell when he was born, as would the death certificate.
The Social Security Death Index will most likely not have it. There are a lot missing prior to 1970's.
An interesting question on Social Security numbers, though. They started to be handed out in 11/1936 according to Wikipedia.
Social Security number - Wikipedia, the free encyclopedia
If that is true, how come there are people who died in as early as 1900 who have SS#'s???
If you go to rootsweb and put in just the death year of 1900 with no names, you come up with 88 people who have SS#s. How can that be? Are they just errors where Social Security put in the wrong date or is there another explanation of how someone can have a SS# who died 36 years before they were issued? Sorry to diverge, but Nutley's post got me thinking.
If you have the SS# for this person and he is not on the SSDI, you can always contact Social Security, pay the $27 or so, and get a copy of the application he filled out and ask for his date of death.
Anyways, those are some ideas of how to get your information.