The quick answer to the question in your title is yes, there is nothing stopping you from relicensing software - only what the current licensees think, paying for something they could have got for free ;)
Having said that, you could always give additional rights under a commercial license. As the copyright holder of your software, you can license it (including dual-licensing) as you see fit. MySQL is a popular example of this: for those who don't wish to follow the GPL's requirements, they can pay for a commercial license.
However, in regards to your question about which license to apply, if you're restricting the license to be for non-commercial research use only, it is not open source.
From point 6 of The Open Source Definition
No Discrimination Against Fields of Endeavor The license must not restrict anyone from making use of the program in a specific field of endeavor. For example, it may not restrict the program from being used in a business, or from being used for genetic research.
Or similarly, freedom 0 of the The Free Software Definition:
The freedom to run the program as you wish, for any purpose
Thus, there is no free/open source license you could apply to this software, with that restriction - you'd need to have a lawyer draft up a specific license that restricted the use as you wanted to.