The Gift Clause is a fan modification rule on the original Nuzlocke ruleset about keeping gift Pokemon. Employing the Gift Clause allows the player to obtain "gift" Pokemon to use in their team, whether it's being obtained from an Nintendo event or from a NPC.
Gift Pokemon are usually received in cities, where random encounters usually don't happen. However the city may be considered a new area, and the gift the first encounter within that area. Following this line of thought, accepting a gift in a city will prevent the player from using a fishing rod or surfing to obtain a Pokémon.
The reasoning for this clause isn't specified, and is different for everyone who uses it. The most common reasoning is simply that the player wants less of a challenge, and since gift Pokemon may include rare or legendary Pokémon in such an effort-free way, employing the Gift Clause will lower the challenge of a Nuzlocke run.
The legitimacy of this clause varies. Hardcore Nuzlocke users would most likely disagree with this clause entirely. More inexperienced Nuzlocke users would most likely allow it. People in the middle might allow it, but only for non-legendary Pokémon. As such the legitimacy of this clause is completely up to the user, however players choosing to eschew this clause are more likely to be accepted as a proper Nuzlocke user.