The Yankees have had the luxury of having an excellent first baseman for a very, very long time. Don Mattingly became the full-time Yankee first baseman in 1984, followed by Tino Martinez immediately after his retirement, Jason Giambi right after Tino left, and then Mark Teixeira immediately after Giambi's departure. That's insane. With no void, the Yankees managed to replace their star first baseman with another star first baseman each and every time the void was created. Unfortunately, they probably won't do that again.
As we all know, free agency has changed. With new TV money and with the new trend of early extensions, not only is free agent talent hard to come by, it's becoming harder to compete as more teams are willing to hand out larger and larger contracts. It's pretty unlikely that a star first baseman will go on the market after Teixeira's departure, so the Yankees will have to get creative in filling the void. As we all know, Teixeira is breaking down. After his major surgery last year, he's been banged up pretty frequently and it's obvious that he's no longer an everyday player. He's great when he's on the field, it's just that health will certainly be an issue until his contract expires.
Replacing Mark Teixeira is a two-pronged approach: the team has to address the issue of a backup first baseman while he's still playing, and then they will have to find a long-term replacement once he leaves. Neither will be an easy task. After this season, the only first basemen of note that will become free agents are: J.P. Arencibia, Mark Reynolds, and Lyle Overbay. I would pass on all of them except for Reynolds, except for the fact that his contract demands will most likely be much higher after a decent season in Milwaukee. The Yankees have the money to absorb such a short term deal, and Reynolds would be ideal in providing near-replacement level performance in Teixeira's stead. WIth Teixeira projected to produce 1.7 and 1.3 WARP in 2015 and 2016, respectively (each over nearly a half season's worth of PA), a player like Reynolds would provide passable performance and the total sum would be an average first baseman. I think using players that have never played first base before as default backup first basemen is an experiment that should be left in the past, especially with the knowledge that Teixeira is definitely breaking down. That could be Reynolds, or it could be a myriad of minor-league free agent signings.
The long-term replacement is obviously the most complicated, and I don't think there's even an answer to this. The best estimation as to how the Yankees will replace Mark Teixeira is either through: the 2017 free agent crop and pick the least of all evils; or, try to use the best hitter in their minor league system as their first baseman. Neither of these options are inspiring. The current first basemen free agent class for 2017 consists of: Justin Morneau, Brandon Moss, Justin Smoak, Logan Morrison, James Loney, Edwin Encarnacion, and Ike Davis. Yuck. If the Yankees want to sign a younger player, there are mediocre options like Davis, Smoak, or Morrison. If they go with proven veterans, they'll likely have to go with an older player like Loney, Moss, or Encarnacion. They would probably be decent, but age is certainly not on their side.
The internal options are underwhelming, to say the least. The best bet, just hitting wise, is Aaron Judge, but I doubt they want to move him from the outfield right now. Unless first base is a total black hole, it would not be wise to put a rock at first base, even if they're hitting well. Eric Jagielo would be another non-first base option, and he would be the most likely to move from position as scouts have recently questioned his range at the hot corner. But if his hitting does not develop appropriately (many scouts label it as 55-60 on a 20-80 scale), then his bat would be a waste at first base. Then, there is of course Greg Bird. Greg Bird is the only and best first base prospect in the Yankees' system, and would be the ideal choice to succeed Teixeira if everything worked out. Between High-A and Double-A this season, Bird has hit .277/.379/.471 with 11 home runs in 86 games. According to PECOTA projections, that translates to .241/.348/.418 with 21 home runs in 2017. I would take that in a heart beat.
The future of first base is a mystery. There won't be a knight in shining armor to save the position as has been the case so many times before; Yankees fans have taken that for granted for nearly 30 years. If I had my dream, Greg Bird would be this person. But if that does not work out and their system fails to produce, the position may become a revolving door for the foreseeable future.