You name a card during the resolution of this spell, at the same time it is being put into play. Once you name the card, there is no chance for any player to respond to this choice before the "can't be played" effect takes hold.
The naming of the card is upon resolution, whether you name it before you put it on the table or after you put it on the table is up to you I guess though the theoretical best would to be to name the card while slamming your mage on the table. =P
Anyway from StarCityGames Virtual Judge
Q: Can Stifle counter Meddling Mage's ability? If yes, can I first know which card he wants to prevent from being played, then Stifle the ability? If not, what kind of ability is this? A static ability?
Can the same be done to Flametongue Kavu, waiting for a target and then Stifling it?
A: Stifle cannot affect Meddling Mage's ability, but it can counter Flametongue Kavu's. The difference is that the Mage's ability is worded "as ~this~ comes into play..." as opposed to "when ~this~ comes into play." The former is part of the resolution of the spell or ability that puts it into play; the latter is a triggered ability.
I would agree and say it isn't a triggered ability as well.