Well, it certainly got its name from a hot trade, and became a byword for a hot-house of another sort…

Limehouse gets its name from the lime oasts or kilns which existed in the area from at least the 14th century and processed the supplies of chalk brought from Kent. It owes its importance however to the links with the sea and growth of London as a commercial centre. In Elizabethan times many famous seaman lived in Limehouse including Sir Humphrey Gilbert, the explorer, and William Borough and Sir Henry Palmer, both Controllers of the Navy. In the early 17th century about half the population of over 2,000 were mariners and a hundred years later when the population had increased to 7,000 Limehouse was considered to be the easternmost part of London. By the 19th century the growth in housing and population and the diminishing importance of the riverside had caused the independent maritime character of Limehouse to disappear. A colony of between 300 and 400 Chinese settled around Pennyfields and Limehouse Causeway about 1890. They were originally seamen from the Blue Funnel Line and the activities of a few opium dealers and gamblers brought some notoriety to the area, although it was much exaggerated.

http://www.eolfhs.org.uk/parish/limehouse.htm