strip mine
Player 2 sacrifices strip mine targeting island
Player 1 in response casts daze returning the island to his hand ... daze fizzles
Daze says "Counter something unless they pay 1"
player 2 didn't play a spell (daze target) so you can't play Daze.
you can, however, float a Blue Mana for say, Brainstorm, but the alternative casting cost of Daze requires bouncing an Island when you play Daze, which is response to a spell (or an ability on the stack, where a spell is already liikely is...)
No matter what, Daze doesn't counter Strip Mine's activated abilities.
you can "Daze" your Island by casting another spell (Brainstorm,
Gush, Disenchant, etc) and pay the 1 estra for Daze to save your Island from said Strip Mine.