What makes statistical distributions so unique? I am going to start this question with a definition.  

Definition:  If $Z\sim \mathcal{N}(0,1)$ and $U \sim \chi_{n}^2$, and $Z$ and $U$ are independent, then the distribution of $$\frac{Z}{\sqrt{\frac{U}{n}}}$$
  is called the $t$ distribution with $n$ degrees of freedom.  

My question , which may sound strange, is, why is this so special?  Why can't anyone just come up with a distribution which is some combination of other random variables and name it after themselves?  
 A: Naming something after yourself maybe considered inappropriate. If the distribution that you discovered is important and useful enough then probably other researchers will already name if after you. Regarding the first question -
"Why can't anyone just come up with a distribution which is some combination of other random variables?"  
You can, but the question is $(1)$ why is your distribution important and $(2)$ what it can be used for? 
The fact that there are around $20$ (or so) "important" distribution is because they are very useful. For example, the normal distribution comes up almost everywhere due to the Central Limit Theorem, Student's $t$ distribution is the bread and butter of basic statistical hypothesis testing (and probably the most used and abused distribution in the medical research), Poisson distribution is the basic counting process that happens to be super useful and versatile in modelling real-world applications, and so on. That is, coming up with a new distribution shouldn't be that hard, coming up with a new analytically convenient distribution will be much harder and coming up with a new analytically convenient and useful distribution is very hard. Therefore, it is unlikely that new distributions, that I'm sure being invented and discovered pretty often, will become popular and widely used.      
