I plan to give my child a new last name for this reason so he/she could have unique google hit. But I am not sure what the legal implications of this are.
I say that because my name is pretty unique -- most hits on my name are relevant to me -- and I have to be very careful about anything I make public. Even more so than most probably have to be.
Fortunately, I'm a pretty boring person and it's not been much a problem.
The bad case IMO is sharing an unusual but not unique pairing with someone who you really don't want to be confused with. Back pre-web, a friend of mine in NYC shared a name with someone who was involved in a fairly serious local baseball scandal. My friend received everything up to and including death threats on his voicemail.
[1] According to "deep web" searches, not entirely but the one or two other cases have no web presence and may well be deceased.
http://www.legislation.gov.uk/uksi/1987/2088/regulation/9/ma...
In the UK, there is no legal formality around changing one's name (you can change it by deed poll if you want a piece of paper, but you change your name by simply using the new name), so they could change it themselves anyway whenever they feel like it.