I would use the name you want. You don't have to tell closer family that it was chosen because of the relative if you don't want to. We deliberately didn't chose a family name (mostly because we didn't want to inflict Ernie and Eric on the poor boys ) but chose a name we both liked. As it happened it turned out to be my great-grandfathers name which please my maternal side no end. We just didn't bother mentioning it to the other sides of the family!
Oh, sorry, a point about the childless thing - my second cousin used my grandma's name (ie her great-great aunt). I was slightly put out (not cross with her, though) as it was my prefered name for a (then mythical) dd. I do think if you are using a name that others might want to use, a bit of checking might be politic (although obviously, nothing to stop the same name being used twice in an extended family)
Hi my sister heroically attended dd2's birth to assist DH (turns out definitely necessary as I was whisked away to theatre and they had to dress and feed her without me!) We said she could choose the middle name but she couldn't decide so her middle name is same as my sisters. She was well chuffed. Was worrying about same ish thing though as dd1 just has a middle name we liked and if we ever have a ds his middle name will be same as my Dad. Worrying my Mum will feel left out!!