What is the word when people come up with the same idea independently

As @Dan Bron points out in his comment, multiple discovery fits the bill perfectly.

Other terms that are sometimes used are simultaneous invention or independent invention. Frequency of occurrence in Google Search are:

    Multiple discovery:     26,600    
    Simultaneous invention: 12,100    
    Independent invention:  69,900 

In Google books:

    Multiple discovery:      4,780
    Simultaneous invention:  4,460
    Independent invention:  33,500

See this Ngram. The figures for independent invention are possibly skewed because this term is used in patent law.

If you're looking for a better, more cool name, synchronicity might work:

the coincidental occurrence of events and especially psychic events (as similar thoughts in widely separated persons or a mental image of an unexpected event before it happens) that seem related but are not explained by conventional mechanisms of causality —used especially in the psychology of C. G. Jung

For added coolness (depending on your age :-)) it's also the title of a 1983 album by The Police.

The idiomatic way to describe just this kind of situation is by saying that the idea was (floating) in the air. And unlike most other suggestions on this page, this particular metaphor is not limited to English to boot.