According to Tyler Kepner of the NY Times, Mark Phillips was the key to this deal.

The Yankees also received Mark Phillips, a 21-year-old left-hander they considered the Padres' best pitching prospect.
The Yankees, who owed White $5 million this season in the last year of his contract, must pay Trammell $7.25 million for the next two years. But they save money this season and add a strong arm to a farm system short on premium prospects.
"The key component in this deal, for us, is Mark Phillips," General Manager Brian Cashman said. "That's not to put any extra pressure on him, but adding an extra power arm to our system was important."
When the Padres took Phillips with the ninth overall pick in the 2000 draft, they gave him a $2.275 million signing bonus. That is roughly the difference between the guaranteed money owed White and Trammell, and Cashman said the Yankees valued Phillips at least that much.