Beason certainly helped solidify the middle of the Giants' defense after he joined Big Blue partway though 2013. Coaches spoke often about how he stepped in as a team leader from the get-go on a floundering team.
Beason, given his age and injury issues, isn't likely to break the bank. That the Giants are focusing their efforts on re-signing him is an indictment of their depleted linebacker corps and also an indication where they sit with their other soon-to-be free agents.
Nicks is all but gone and the Giants appear willing to see how the market shapes up before trying to bring back Tuck and/or Joseph.
The Giants could still bring back the fan favorite, but they'd rather do it at a cheaper price than the restricted tender would call for Hynoski.