Monday is the first day NFL teams can designate franchise or transition tags in an effort to keep players they deem most valuable off the market. (They have until March 3 to decide whether to do it.) Teams don't have to use the designations, and most don't. But if they do, they have the choice to use the exclusive franchise tender, the non-exclusive franchise tender or the transition player designation, which no one uses anymore. A team can pick only one player, at most, on whom to use one of these designations per year. A brief explanation of the differences between the three can be found here.
What we want to figure out here, however, is the likelihood that the New York Giants will use one of these designations this year. We'll ignore the transition tag, because it's outdated and no longer used, and instead focus on the franchise tag. Historically, the Giants have used the franchise player designation as a means of holding a player in place because they believed they were making good progress with that player on a long-term deal and didn't want him to hit the market. Example: Two years ago, nearing completion on a new deal with punter Steve Weatherford, they used the franchise tag on him at the deadline but shortly after announced a long-term deal that superseded it.
They could do that this year if they find themselves in a good position in negotiations with one of their many free agents. Linebacker Jon Beason is a player they'd like to bring back and with whom they've discussed a long-term deal. The franchise-tag salary for linebackers is likely to be more than $10 million, though, so they'd have to feel confident about their chances of signing Beason long-term (they'd have until July 15 to do so) if they were to risk paying him that much on a one-year deal.
Defensive tackle Linval Joseph is another pending free agent who's a candidate for the franchise tag, which for his position should be around $9 million. If they're doing a long-term deal with Joseph before the market opens, it likely would be for less than that, but they could conceivably risk carrying him at that number. I do not believe they will use the franchise tag on wide receiver Hakeem Nicks, for whom the tag could be worth more than $11 million.
It's possible that the strongest Giants candidate for the tag could be kicker Josh Brown, since using the tag for kickers and punters is generally a palatable $3 million or so. The Giants like Brown and could try to sign him to a multi-year deal, using the tag in the meantime as they did with Weatherford.