The Bengals are very deep across the board, which makes this hypothetical very interesting.
Assuming he was healthy, the Bengals could really use a play making middle linebacker like Paul Posluszny - the guy is a tackling machine. Bengals fans are less than enamored with Rey Maualuga, and putting Posluszny next to Burfict would be a pretty sight. However, at age 30, I don't think Posluszny would be the best pickup long term.
Therefore, the guy I would take would be second year left tackle, Luke Joeckel. I know he hasn't lived up to his draft status at this point in his career, but the Bengals wouldn't need him immediately. They could have him sit for a year (or have him play left guard) while learning from Pro Bowl left tackle Andrew Whitworth.
Whitworth will be 33 before the end of the season and the Bengals would be well served to start looking for his replacement. Joeckel is young, athletic and very talented, with tons of potential as a franchise left tackle. While Whitworth is still a good tackle, he is an even better human being and would be the perfect mentor for a guy like Joeckel.
I would sit Joeckel for the year--or possibly play him at left guard next to Whitworth--and then move Whitworth to left guard (where he is actually better) and allow Joeckel to take over Dalton's blindside. Putting Whitworth at guard next to Joeckel would help Joeckel's transition, extend Whitworth's career, and keep the Bengals' leader on the team for the next few years.