If I'm remembering correctly, according to Atkins and Rundell (2008) , many projects build their own software, but there are also commercial packages such as TshwaneLex (http://tshwanedje.com/tshwanelex/).
Atkins and Rundell (2008) go into some detail about the requirements of such software, so you may find their book helpful. They mention that there are two types of software which are needed for any lexicography project: what they call a Corpus Query System (CQS) and what they call a Dictionary Writing System (DWS). The point of a CQS is to access evidence on the basis of which dictionary entries can be written; in that sense, their needs are very similar to those of standard corpus linguistic software* (which include the ability to draw up concordances, as well as other features). The point of the DWS is to offer an interface for generating and storing what will make up the content of the dictionary.
 Atkins, B.T. Sue and Michael Rundell (2008) The Oxford Guide to Practical Lexicography.
*an example of corpus linguistic software is AntConc, which is free, though probably not powerful enough for a lexicography project. You can see a list of commonly used corpus linguistics software here: http://www.corpora4learning.net/resources/materials.html#5.