Would you rather have a common first name and an uncommon last name or vice versa?

both.

I grew up with a name that is very very common in Italy, but vitually unheard of in the US where I lived mostly. My first name was very common in a few European countries, but a rare classic.

So, both were normal, but rarely heard names and I loved it!

That is what I will be doing for my children. Their names are quite popular worldwide, but not so much where we will live.

I have an uncommon first name (although it's more common the last 15 years or so) and an average not either common or uncommon last name. I like it that way.

uncommon last name