Question: To my understanding: A separable extension $K/F$ is one in which the minimal polynomial of every $\alpha\in K$ has no multiple roots. A normal extension $K/F$ is one in which some polynomial $f\in F[X]$ splits over $K$.

Is there any reason why these concepts are so called, or are they just arbitrary names?

Answer: 
I always thought, but this is not a historic answer, that "separable" comes from the fact that the roots are "separate" in the sense that there are no repeated roots. 

This is in line what I just found when searching [Keith Conrad in his lecture notes writes](http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/separable1.pdf): 

> 
> The term 'separable' comes from distinctness of the roots: they are separate in the sense that there are no multiple roots.
> 
> 
> 

For "normal" I would say it is pretty arbitrary except that the name should somehow convey that these are the good/well-behaved/regular extension. Sometimes a normal extension is also call quasi-Galois.
This is very much so a stab in the dark (read: definitely not historical) but sometimes when people say "normal" extension they mean Galois. Similarly, over perfect fields (all extensions are separable; e.g. char 0) normal extensions <-> Galois extensions.

If we have a Galois extension $K/F$, $F$ perfect (is this necessary?), normal subgroups of $\operatorname{Gal}(K/F)$ correspond to normal subextensions of $K/F$.