There are at least three sources of confusion here: quotes don't (generally) nest, $variable references are expanded by the shell even if they're in double-quotes, and variable references are resolved before var=value assignments are done.
Let me look at the second problem first. Here's an interactive example showing the effect:
$ NAME=Gordon
$ bash -c "NAME=me echo $NAME"
Gordon
Here, the outer (interactive) shell expanded $NAME before passing it to bash -c, so the command essentially became bash -c "NAME=me echo Gordon". There are several ways to avoid this: you can escape the $ to remove its normal effect (but the escape gets removed, so the inner shell will see it and apply it normally), or use single-quotes instead of double (which remove the special effect of all characters, except for another single-quote which ends the single-quoted string). So let's try those:
$ bash -c "NAME=me echo \$NAME"
$ bash -c 'NAME=me echo $NAME'
(You can't really see it, but there's a blank line after the second command as well, because it didn't print anything either.) What happened here is that the inner shell (the one created by bash -c) indeed got the command NAME=me echo $NAME, but when executing it expands $NAME first (giving nothing, because it's not defined in that shell), and then executes NAME=me echo which runs the echo command with NAME set to "me" in its environment. Let's try that interactively:
$ NAME=me echo $NAME
Gordon
(Remember that I set NAME=Gordon in my interactive shell earlier.) To get the intended effect, you'd need to set NAME and then as a separate command use it in an echo command:
$ bash -c "NAME=me; echo \$NAME"
me
$ bash -c 'NAME=me; echo $NAME'
me
Ok, with that out of the way let's move on to the original question about quoting. As I said, quotes don't (generally) nest. To understand what's going on, let's analyze some of the example commands. You can get a better idea how the shell interprets things by using set -x, which makes the shell print each command's equivalent just before it's executed:
$ set -x
$ bash -c "NAME=me echo "Hi $NAME""
+ bash -c 'NAME=me echo Hi' Gordon
Hi
What happened here is that the shell parsed "NAME=me echo "Hi as a double-quoted string immediately followed by two unquoted characters; since there's no gap between them, they get merged into a single argument to bash -c. It may seem a little weird having only part of an argument quoted, but it's actually entirely normal in shell syntax. It's even normal to have part of a single argument be unquoted, part single-quoted, part double-quoted, and even part in ANSI-C mode ($'ANSI-c-escaped stuff goes here').
With set -x, bash will print something equivalent to the command being executed. All of these commands are equivalent in shell syntax:
bash -c "NAME=me echo "Hi Gordon
bash -c "NAME=me echo Hi" Gordon
bash -c 'NAME=me echo Hi' Gordon
bash -c NAME=me\ echo\ Hi Gordon
bash -c NAME=me' 'echo' 'Hi Gordon
bash -c 'NAME=me'\ "echo Hi" Gordon
...and lots more. With set -x, bash will print one of these equivalents, and it just happens to choose the one with single-quotes around the entire argument.
Just for completeness, what happened to $NAME""? It's treated as an unquoted variable reference (which expands to Gordon) immediately followed by a zero-length double-quoted string, which doesn't do anything at all.
But... why does that just print "Hi"? Well, bash -c treats the next argument as a command to run, and any further arguments as the argument vector ($0, $1, etc) for that command's environment. Here's an illustration:
$ bash -c 'echo "Args: $0 $1 $2"' zeroth first second third
+ bash -c 'echo "Args: $0 $1 $2"' zeroth first second third
Args: zeroth first second
("third" doesn't get printed because the command doesn't print $3.)
Thus, when you run bash -c 'NAME=me echo Hi' Gordon, it executes NAME=me echo Hi with $0 set to "Gordon".
Ok, here's the last example I'll look at:
$ bash -c "NAME=me echo Hi "" $NAME"
+ bash -c 'NAME=me echo Hi Gordon'
Hi Gordon
What's happening here is that there's a double-quoted section "NAME=me echo Hi " immediately followed by another one, " $NAME", so they get merged into a single long argument (which happens to contain two spaces in a row -- one part of the first quoted section, one part of the second). Essentially, the "" in the middle ends one double-quotes section and immediately starts another, thus having no overall effect. And again, the shell decided to print a single-quoted equivalent rather than any of the various other possible equivalents.
So how do we actually get this to work right? Here's what I'd actually recommend:
$ bash -c 'NAME=me; echo "Hi $NAME"'
+ bash -c 'NAME=me; echo "Hi $NAME"'
Hi me
Since the entire command string is in single-quotes, none of these problems occur. The double-quotes are just normal characters being passed as part of the argument (so double-quotes sort of nest inside single-quotes -- and vice versa -- but it's really just 'cause they're ignored), and the $ doesn't get its special meaning to the outer shell either. Oh, and the ; makes this two separate commands, so the NAME=me part can take effect before the echo "$NAME" part uses it.
Another equivalent would be:
$ bash -c "NAME=me; echo \"Hi \$NAME\""
+ bash -c 'NAME=me; echo "Hi $NAME"'
Hi me
Here the escapes remove the special meanings of the $ and enclosed double-quotes. Note that the shell prints exactly the same thing as last time for its set -x output, indicating that this really is equivalent to the single-quoted version.
