I think the 'Put' in his first name probably refers to 'Putin', which can be seen in some Russian surnames (like the surname of the current Russian president, and the infamous Rasputin). In Rasputin's case, his surname meant "dissolute" or "profligate." So maybe the 'putin' thing is something negative in Russian, but also feel it's something of a Russian adjective.
I've found nothing on the Internet where Putnev is listed as a first name, so your new grandson was probably named after a city or carries a surname as a first name.
I think the best option is to give the boy an English first name which resembles Putnev a bit, like Peter. Another option is to just give the boy the surname your daughter and son-in-law like, and have the boy keep Putnev as his middle name. That would be the best option, in my opinion.