Because Byzantion was very far out of his way.

Xerxes gathered his army at Sardis, the main Persian administrative centre in Asia Minor, which was located inland from modern Izmir. From there, Xerxes would have marched down to the coast and north to get to the straits where Asia and Europe are closest together. A glance at Google Maps will tell you that this would have led him directly to the Hellespont. A crossing there would also have put him well on the way west, which was the direction he'd have to take to get to mainland Greece.

Had he tried to cross at Byzantion, on the other hand, he'd have had to march hundreds of kilometers back east along the Propontis, only to have to double back and march west again once he'd crossed over.

Herodotos also suggests that the difference in width of the two straits was not as great at the time as it is today. He claims that the bridge built at Byzantion by Darius for the invasion of Thrace and Skythia in 507 BC (where it made sense to cross in the north, since Darius was marching north) had to cover a width of about 4 stades, c. 800m (Hdt. 4.85.3). The bridge that Xerxes ordered to be built at Abydos, north of modern Çannakale, spanned a strait of about 7 stades, c. 1400m (Hdt. 7.33-34). The extra length was not so great as to make the operation a senseless waste, and the project was achieved despite adverse weather through the use of pontoon hulls and other feats of engineering.

Besides these practical reasons, we also shouldn't lose sight of the importance of Persian royal ideology. The bridge-building operation was not just a solution to a practical problem, but a display of the power, resources and ingenuity commanded by the Great King. Xerxes would have been driven by the need to outdo his predecessor, and would not have been satisfied with merely matching Darius' achievement of bridging the Bosporus.