Why the word "mind" can be used as a verb, synonym of "pay attention to"? It has the same etymology of the "mind" (centre of thought, feelings, brain) noun? When it is better to use "mind" in place of "pay attention to" (or similar phrases)?
The verb, "to mind" is descended from the Old English noun "gemynd", which means "thought", "memory", "thinking", "commemoration", and "intention". This word itself is descended from the hypothetical Proto-Indo-European word "*men" meaning "to think", "to remember", or "to have one's mind aroused".
The most interesting thing (for me) is that this word has cognates in Sanskrit, Latin, Ancient Greek and Russian.
I guess the word would have originally been used in the sense "to remember". E.g. when one is saying "mind the kids!" what they really mean is "don't forget the kids!". The word, with usage, would have eventually gained its meaning of "to pay attention to", simply through its continual usage in that sense. It gained this particular meaning ("to take care of", "look after") round about the 1690s.
Seeing as "mind" has so many meanings to do with thinking, remembering, as well as the whole idea of spiritual embodiment of one's thoughts, it only makes sense that it would have so many uses with different flavours of meaning today.
As for exactly how it happened; that would require a lot more research, and possibly a time machine.
For usage, I don't think it makes much difference whether someone were to say/write "pay attention to the kids" or "mind the kids". The latter quote is a lot shorter, but also has a slightly different meaning.
Warning: Opinion follows: "Pay attention" implies that you need to watch them, but doesn't necessarily mean you must look after them as well - although it usually would. "Mind the kids", implies that you must watch the kids and keep them out of danger - though if anyone were watching kids, and didn't take care to keep them out of danger, they would be a right drongo.
"Pay attention to..." can also mean "Pay especial attention to...", for instance, in the sentence
Mind the kids, and pay attention to Sally.
It is obvious that the listener should look after all the kids, with special care regarding Sally, as she may be prone to being more naughty or adventerous, and likely to get into trouble/injure herself.
To put the two phrases in the other order:
Pay attention to the kids, and mind Sally especially.
Requires the extra adverb to bring the same meaning to the table.
"Mind" is short for "keep in mind." In this context, it is therefore synonymous with (but not the same as) "pay attention to."
Let's start with "pay attention". The phrase is very old and dates back to a time when "pay" meant "give" as stated here: What is the origin of the phrase "pay attention"?. Actually, though, beyond what this answer states it's also important to note that the historical meaning of "pay" also meant to pacify, or to be pleasing or satisfactory to someone. While these meanings are obsolete, my observation is that hints of them still exists in phrases like pay attention - so the meaning ranges from "give attention" to "give attention in a way that is pleasing or satisfactory to [me or someone else]." See: http://www.grammarphobia.com/blog/2015/03/pay-attention.html
There's also a range of meanings for mind. One end of the range is simply to "give attention", so both words start their ranges at exactly the same place. But the range of meaning for mind can extend all the way to heed, or even to obey. As noted on Thesaurus.com, "Mind is that part of a human being that thinks, feels, and wills" so the range of meaning when someone says "mind" can be, on one hand, to think about something (which would be commensurate with giving it attention) but could also mean to feel something, or to be of one will with something, which is far beyond merely giving it attention.
So if I say "pay attention" I can imply a range of meanings from: a) give attention to b) give attention in a way that is pleasing or satisfactory
If I say "mind" I can imply a range of meanings from: a) give attention to b) give attention in a way that is pleasing or satisfactory or even further to c) be of one feeling to d) be of one will
Understanding the above, if I want you to give something temporary focus, I can use either: "this part is important so pay attention to what I'm about to say" or "this part is important so mind what I'm saying next"
However, if I want you to give a more stronger or permanent connection to something, pay attention is no longer satisfactory and I need to use mind: "mind your mother while I'm away" "mind the words of Dr. King because violence will solve nothing in this situation"
Final comments. There was an interesting example in another answer, making the argument that if a person used the word order "mind" (a group) and "pay attention to" (a smaller element of that group) that "especially" would be understood and not need to be stated specifically, whereas in the reverse, (the exact example was "Pay attention to the kids, and mind Sally especially") the phrase needed that clarifying word. What I think the other answer missed was that the main reason that extra word helps the phrase sounds more natural in the example is that "mind" is so much of a broader word. For instance, if one said "while you're home from college pay attention to what everyone in the family has to say and mind your father" it would be interpreted to go beyond merely giving your father attention and indeed heed/obey him, unless one added "especially" to indicate one merely meant "special attention". Therefore, I don't agree with the conclusion in the other answer, although I will say the other answer started down the right road, noting that "mind the kids" has a broader range of meaning than "pay attention to the kids" - I do agree with that. I think "mind the kids" includes the meaning "give attention to and be of one feeling with the kids and heed the needs of the kids" whereas "pay attention to the kids" means only to "give attention to the kids".