Battle of Hoover Dam
As this enchantment enters, choose NCR or Legion. • NCR — At the beginning of your end step, return target creature card with mana value 3 or less from your graveyard to the battlefield with a finality counter on it. • Legion — Whenever a creature you control dies, put two +1/+1 counters on target creature you control.
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Official rulings (5) · Scryfall
Finality counters aren’t keyword counters, and a finality counter doesn’t give any abilities to the permanent it’s on. If that permanent loses its abilities and then would go to a graveyard, it will still be exiled instead.
If you somehow control Battle of Hoover Dam and no choice was made for it (perhaps because another permanent on the battlefield became a copy of it), it has neither of the two triggered abilities.
Multiple finality counters on a single permanent are redundant.
Finality counters work on any permanent, not only creatures. If a permanent with a finality counter on it would be put into a graveyard from the battlefield, exile it instead.
Finality counters don’t stop permanents from going to zones other than the graveyard from the battlefield. For example, if a permanent with a finality counter on it would be put into its owner’s hand from the battlefield, it does so normally.