Mainframe rehosting is about replacing the whole mainframe with one or multiple Linux boxes, or at times, move portions of the application landscape from the mainframe to Linux. Thereby you basically keep many of the hitherto used development- and runtime-environment, like COBOL, DB2, IMS, CICS, etc. The goal in mainframe rehosting is to dramatically reduce costs. If you are dauntless you can also move to Windows.
The point is that you do not rewrite your applications but rather just move your applications, i.e., you recompile your applications on new hardware. So all accumulated experiences with the software is fully preserved.
In 2008 I started writing a paper on mainframe rehosting, see Mainframe Rehosting.
During 2008-2011 the paper has been revised somewhat. The paper is in German.