"Pay attention" vs. "give attention"
Solution 1:
As far as I can tell, pay is used when there is no possessive adjective (e.g. my, his, etc) before the attention:
I didn't pay him much attention.
Give is used when there is such an adjective:
All of us were supposed to give Chris our undivided attention.
It is interesting that give attention (without the possessive) hasn't always been uncommon, though it's always lagged behind by a factor of at least two:
As you see, pay attention currently leads by ten times or more.