If I play Counter Strike in response to my opponent’s Elder Dragon’s play effect, will the chosen unit still die?

No, the unit protected by Counter Strike will not die.

Elder Dragon’s passive ability changes the definition for what is considered lethal damage for a unit, as long as that unit has damage marked by Elder Dragon’s controller on it. However, this doesn’t change the actual amount of damage being dealt. Elder Dragon deals 1 damage to up to one unit at each location—if one of those units has a prevent effect applied to it, that prevent effect will replace as much of that damage as it can. In the case of Counter Strike, it will replace all of the damage. If the damage is entirely prevented, it is never dealt per rule 437.2.a., and thus there is no opportunity for the passive ability of Elder Dragon to apply to that unit.

Rule 437.2.a.: The damage being dealt as a result of Preventing can never be less than 0, but can be 0. This is equivalent to not dealing damage.