An owner of a fantasy football team who doesn't use their waivers to pick up a player and instead they wait until the early morning for certain players to clear waivers. For context, when a player is dropped in fantasy football, they go on to "waivers." The waiver period typically lasts a few days, and during this period, a team cannot pick up a player on waivers without sacrificing their place in the waiver rotation. A team with a higher waiver spot for the week will have preference over another team if both are attempting to claim the same player.
To be a waiver hawk for a specific player means you are attempting to bypass the waiver system on a certain player by waiting until the minute (literally) after a player's waiver period ends in order to not lose your position in the waiver rotation.
There is strategy behind being a waiver hawk for certain players. A team would risk its waiver rotation spot for a player they believe will be a hot commodity and other teams may also try to pick up. However, for players that may be more under the radar, you would rather not waste your spot in the waiver rotation and instead wait until the time has passed for the player to be on waivers, and get him with sacrificing anything.