What is the meaning of the word 'power of death' which Satan was holding as in Heb 2:14.?
The Devil is considered a prince of a kingdom. To be a prince he must have some kind of power that he wields over his subjects. If we said a king has the power of wealth, we mean he has a lot of money and that he exerts a lot of influence by it. In the same way the 'power of sin' or 'power of death' is something one can have and exert influence with. The Devil having introduced sin into the world and through sin death, became a prince of the world having a lot of destructively powerful sin and death in himself. Like a disease, the Devil, with the power of death, causes sin and death to spread through his influence. He according to this power works his desire to destroy and kill. He by his daily temptations increases death in the world, a world under his power as a result of being made subject to death.
The Devil by tempting Adam and Eve brought all men under a power of death
Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all people, because all sinned —(NIV, Romans 5:12)
He is a prince over the kingdom of the world which resulted from death entering into it
As for you, you were dead in your transgressions and sins, in which you used to live when you followed the ways of this world and of the ruler of the kingdom of the air, the spirit who is now at work in those who are disobedient. (NIV, Ephesians 2:1-2)
Although there may be nothing legitimate about his power of sin and death, he nevertheless has great power in his rebellion as the prince of this world and is in that sense like a powerful Lion:
Be alert and of sober mind. Your enemy the devil prowls around like a roaring lion looking for someone to devour. (NIV, 1 Peter 5:8)
Therefore to conclude with a pointer from the obvious beginning, God said the moment Adam ate from the fruit he would die but this could not have been primarily physical as he did not die. Also that they felt ashamed, hid from God and this proves they did die inwardly. The bible does not treat physical death as serious as spiritual death anywhere.
Question Restatement: "Power" seems to be ambiguously translated in this context, leaving one to ask: Does this term imply: Authority (ἐξουσίαν), Divine Might/Miraculous (δύναμιν), or Strength (κράτος).
Heb. 2:14 - Inasmuch then as the children have partaken of flesh and blood, He Himself likewise shared in the same, that through death He might destroy him who had the power of death, that is, the devil,
Heb. 2:14, BYZ - Ἐπεὶ οὖν τὰ παιδία κεκοινώνηκεν σαρκὸς καὶ αἵματος, καὶ αὐτὸς παραπλησίως μετέσχεν τῶν αὐτῶν, ἵνα διὰ τοῦ θανάτου καταργήσῃ τὸν τὸ κράτος ἔχοντα τοῦ θανάτου, τοῦτ’ ἔστιν τὸν διάβολον,
Answer: Power, in this text should be "Strength (from κράτος)." More specifically, this also relates to the "Agony of death" (below). Another valid translation would be: "That he may [render irrelevant/powerless] the one who has the Strength of Death, that is, the Devil.
The Agony of Death:
Consider it the "Death Power" ...
Luke 16:24, NASB - And he cried out and said, ‘Father Abraham, have mercy on me, and send Lazarus so that he may dip the tip of his finger in water and cool off my tongue, for I am in agony in this flame.’
Acts 2:24, NASB - But God raised Him up again, putting an end to the agony of death, since it was impossible for Him to be held in its power.
Death is the physical evidence or manifestation of one that is in the domain of the devil,death are in stages sickness, disease, corruption, decay and its final blow "death" to separate one from the soul and out of the earth.