Ask Mr. Smarty Plants

Double-flowered columbines are not unheard-of, though they're not exactly common either.  You will want to first record as much information about your plant as possible.  Get excellent photographs of not only the flowers, but all of the plant parts from various angles.  Start keeping notes about your plant, when it flowers, if and when it sets fruit, what it's growth habit is like.  Do a lot of web-searching to see if a cultivar like yours plant already existes.  Contact your county Cooperative Extension Service agent to find out if your state's universities have horticultural experts who might assist you.

You will need to propagate your plant.  Columbines (genus, Aquilegia) are usually propagated by seed.  Since this genus is very prone to hybridizing, the seedling offspring of the plant in question are unlikely to have the same characteristics as the parent.  However, Columbines may also be propagated by plant divisions.  Usually, by the time a plant is two or three years old it is ready to be divided.  Increasing a cultivar through propagation by plant division is, of course, a slow process.

The part of your question about naming leads inexorably into legal areas for which we have neither the expertise nor the license to give specific advice. 

Plants new to horticulture can be patented through the US Patent office, but this is a slow, usually expensive, and often frustrating process.  Novices to the world of plant patents would probably be well-advised to hire professionals who specialize in patents and trademarks to assist in the process.

Some who discover or develop new cultivars work with established nurseries or seed companies who already have a lot of expertise in developing and introducing new plants to horticulture.  Common sense would dictate being careful about who you entrusted with your new discovery.  Of course, if you have no desire to receive any remuneration from your discovery and wish only to share it with the world, then the whole process will be simpler.

For naming horticultural plants, The International Code of Nomenclature for Cultivated Plants is your bible.  The website of the International Society for Horticultural Science maintains a listing of International Cultivar Registration Authorities.  These are professional organizations dedicated to the study and promotion of various plant genera and other plant groups.  Surprisingly, it does not appear that there is currently an ICRA for Aquilegia.  However. the ICRA homepage gives instructions on whom to contact when no official registration authority exists.