Usually the way this works is that a researcher discovers some phenomenon and publishes it without giving a name. Then someone else comes along and writes about "Jones' discovery" and before you know it the world talks about the "Jones effect".
Giving things a name yourself is done in some fields - for example, in MRI (magnetic resonance imaging) it is conventional that people who develop a new pulse sequence will give that sequence a "catchy" name (usually some clever abbreviation, never really their own name). So you have names like CAIPIRINHA (which is really the name of a drink, but stands for "Controlled Aliasing In Parallel Imaging Results IN Higher Acceleration").
In the end, the name that the community adopts is the name by which something will be known - so you are free to call it the "teletubby galaxy" but if everybody insists on calling it Andromeda-X42 you will be a lone voice. If you have the good fortune of discovering something new and valuable, name it wisely or be ignored.
Reiterating the point made by Nathaniel: suggesting that a particular phenomenon/constant/equation should be named after you is considered a major faux pas. Others will do it for you - when you try to jump the gun and suggest that it be named after you, you will come across as conceited. I can think of no example of a well known effect that a discoverer named for themselves. Some examples where they did not:
Newton's Laws (he called them "Axiomata sive Leges Motus")
Hooke's Law ("Explaining the power of springing bodies")
Josephson effect (for which he got the Nobel prize... not available for Newton or Hooke, or surely they would have qualified) was described by him (Physics Letters Vol 1, No 7, 1962) as "new effects":
- he didn't say "we present here the Josephson Effect".
The list could go on and an. If you are the greatest, you don't need to say so - others will do it. No offense intended, Mr Ali.
If you are interested interested, there is a follow-up question on a sister site, History of Science and Math: What famous laws were named by their discoverer?