What is another name for London Bridge?

There isn't, it is London Bridge. There has been a bridge on the site for nearly 2,000 years. It connects the City of London to Southwark. The current one was opened in 1973 replacing the John Rennie designed bridge of 1831 which replaced an earlier bridge which was 600 years old. It is included in the famous nursery rhyme "London Bridge is falling down" dating from the 17th century but whose origins may be older. The John Rennie bridge was sold to an Amercan entrnepreneur who rebuilt it in Arizona. There is an urban myth that he mistook it for the more impressive Tower Bridge built in 1894 which has been denied by all involved. A mistake which was however given by an earlier answer and quite a common error by tourists, Tower Bridge is the next one down by the way!