Does My Mother Just Have a Dirty Mind?
--Potential Parents of Peter
Grandma's got it both right and wrong. While it's true that the word "peter" was occasional used in the way she fears in the past, the slang is no longer current today. You could even say it has "petered out." After all, you've never heard that usage, which should be reassuring.
Peter is a classic name that might appeal to you because it's familiar, yet not common. It peaked back in the 1950s, and now sits comfortably in the 200 ranks in popularity. It's traditional, masculine, and solid. That's what people will hear when they hear the name.
So while I'd probably advise against using Dick as a nickname for Richard in this day and age, Peter doesn't pose the same problem. After all, there are still plenty of baby boys being named John, even though we're still using the word "john" in more than one unflattering way. There are boys named Cooper and Tucker who seem to be surviving the potential for rhyming taunts just fine.
If you love the name Peter, use it. There are no tease-proof names, but schoolkids are more accepting of varied names than in back in parents' — and grandparents' — days. (And you can always call him Pete!)