What kind of free program can I use for that?
You can use Adobe DNG converter and select the option to compress the DNGs.
http://www.adobe.com/products/dng/What is the easiest way to add the same name to the jpeg and dng files?
IMO, the best way to rename your pictures is with the date and time. All your pictures (and I mean ALL) will have a unique name, thus you'll never overwrite them by mistake. You can quickly search for a group of pictures if you remember when you took them, for example certain holyday or birthday. Also you can sort them easily in any operative system.
In your case, you can rename both the JPG and DNG by using exiftool which is a free and powerful application to handle photo meta-data. I use the following command line instruction:
C:\apps\exiftool\exiftool * "-FileName<CreateDate" -d %%y%%m%%d_%%H%%M%%S%%%%-c.%%%%e .
This will rename all the files in the current folder with the format YYMMDD_HHMMSS, for example 100731_144047.JPG where the date is 31st of July 2010 and the time 2:47:47pm.
The date and time is extracted from the picture's EXIF data and if you created your DNGs on camera they should have the same date as the JPGs and both will have the same name but different extension.
Have fun.