The death of Frank R. Lautenberg on Monday has left Gov. Chris Christie of New Jersey with the kind of opportunity that politicians usually covet: the chance to give away a seat in the United States Senate. But the decision is fraught with pitfalls, none bigger than having to choose between improving his party’s fortunes in Washington and furthering his own political ambitions at home.

Mr. Christie, a Republican, is up for re-election in November and hoping to secure a huge victory margin, which he could then use to accelerate his drive to present himself as a presidential candidate with broad appeal even in a blue state.

But adding a special election for the Senate seat to the ballot could put Mayor Cory A. Booker of Newark at the top of the Democratic ticket, potentially energizing more Democrats, who already outnumber Republicans in New Jersey by 700,000 registered voters, to come to the polls.

The implications extend beyond New Jersey to Washington, where both parties are maneuvering for strategic advantage in a Senate where even a single vote can derail legislation. The appointment of a Republican as the interim replacement for Mr. Lautenberg, a Democrat, would create immediate complications for the White House and Democrats on Capitol Hill as they try to push through presidential nominations and an overhaul of immigration laws under the constant threat of Republican filibuster.