The thing you call a variable is a label that basically holds the address of the value in memory. When you want to change the value you need to use brackets [] and dereference the address that points to that location. Then you can change the values one by one. For example, lets define a one-byte variable:
v: db 0x00
To change the value you can do
mov byte[v], 0x02
As you can see we specified the size with byte
If we had the following variable:
abc: dw 0x0000
the variable abc would only hold the address of the first byte of the data but the data itself is a word (2 bytes). That is why to change the variable's value we need to do:
mov word[abc], 0xDEAD
which would be equivalent to
mov byte[abc], 0xAD
mov byte[abc+1], 0xDE
Note that the least first byte of the 2-byte value is in the earlier memory address, this is called little-endian order.
A string is essentially a bunch of "bytes" next to each other (it doesn't use little endian). To change a string value one by one you can do:
text: db "Hello World", 0
mov byte [text], 'A' ; Aello World
mov byte [text+1], 'B' ; ABllo World
mov byte [text+2], 'C' ; ABClo World
; and etc 
Also finally we can take a look at your code:
text db "Hello, World!",10
   
mov rax , "He"
mov [text], rax
syscall
This is not valid (as pointed out by @vitsoft) because you are putting "He" inside of rax before calling syscall which uses rax to determine what it's gonna do.
As a matter of fact this line of code
mov word [text], "He"
is perfectly valid. I don't know why you couldn't get that to work. "He" is essentially resolved to 0x6548 and you do a normal mov as a word. As I mentioned before because of the little-endian order for words, 0x48 ('H') will be placed in the first byte of text which is already "H" and similarly 0x65 ('e') will be placed in the second byte of text which is already "e".
Edit:
Lets say you don't know the length of a string which you want to copy to another string/location. In that case you should loop over that string and do the changes one by one. I will leave a sample code here which you would need to fix and adapt:
start:
    xor ecx, ecx ; initialize some variable to keep count
.loop:
    mov al, byte [other + rcx] ; get the nth character of other.
    cmp al, 0x00 ; if we reached the end of the string
    je endLoop ; end the function
    mov [text + rcx], al ; write the nth character of other to nth position of text
    inc ecx ; increase counter
    jmp .loop ; loop
endLoop:
    ret
text: db "Hello World", 0
other: db "ABC", 0