Using sscanf to read strings

You must allocate space for your strings, e.g:

char* old_name = malloc(128); 
char* new_name = malloc(128);

Or using arrays

char old_name[128] = {0};
char new_name[128] = {0};

In case of malloc you also have to free the space before the end of your program.

free(old_name);
free(new_name);

Updated:...

The other answers provide good methods of creating memory as well as how to read the example input into buffers. There are two additional items that may help:

1) You expressed that you want to ignore the quotation marks too.
2) Reading first & last names when separated with space. (example input is not)

As @Joachim points out, because scanf and family stop scanning on a space with the %s format specifier, a name that includes a space such as "firstname lastname" will not be read in completely. There are several ways to address this. Here are two:

Method 1: tokenizing your input.
Tokenizing a string breaks it into sections separated by delimiters. Your string input examples for instance are separated by at least 3 usable delimiters: space: " ", double quote: ", and newline: \n characters. fgets() and strtok() can be used to read in the desired content while at the same time strip off any undesired characters. If done correctly, this method can preserve the content (even spaces) while removing delimiters such as ". A very simple example of the concept below includes the following steps:
1) reading stdin into a line buffer with fgets(...)
2) parse the input using strtok(...).

Note: This is an illustrative, bare-bones implementation, sequentially coded to match your input examples (with spaces) and includes none of the error checking/handling that would normally be included.

int main(void)
{
    char line[128];
    char delim[] = {"\n\""};//parse using only newline and double quote
    char *tok;

    char letter;
    char old_name[64];  // Space for 63 characters plus string terminator
    char new_name[64];

    fgets(line, 128, stdin);
    tok = strtok(line, delim);     //consume 1st " and get token 1 
    if(tok) letter = tok[0];       //assign letter
    tok = strtok(NULL, delim);     //consume 2nd " and get token 2
    if(tok) strcpy(old_name, tok); //copy tok to old name
    tok = strtok(NULL, delim);     //consume 3rd " throw away token 3
    tok = strtok(NULL, delim);     //consume 4th " and get token 4
    if(tok) strcpy(new_name, tok); //copy tok to new name

    printf("%c %s %s\n", letter, old_name, new_name);


    return 0;
}

Note: as written, this example (as do most strtok(...) implementations) require very narrowly defined input. In this case input must be no longer than 127 characters, comprised of a single character followed by space(s) then a double quoted string followed by more space(s) then another double quoted string, as defined by your example:

N "OldName" "NewName"

The following input will also work in the above example:

N    "old name"             "new name"

N "old      name" "new        name"

Note also about this example, some consider strtok() broken, while others suggest avoiding its use. I suggest using it sparingly, and only in single threaded applications.

Method 2: walking the string.
A C string is just an array of char terminated with a NULL character. By selectively copying some characters into another string, while bypassing the one you do not want (such as the "), you can effectively strip unwanted characters from your input. Here is an example function that will do this:

 char * strip_ch(char *str, char ch) 
 {

    char *from, *to;
    char *dup = strdup(str);//make a copy of input

    if(dup)
    {
        from = to = dup;//set working pointers equal to pointer to input
        for (from; *from != '\0'; from++)//walk through input string 
        {
            *to = *from;//set destination pointer to original pointer 
            if (*to != ch) to++;//test - increment only if not char to strip
                                //otherwise, leave it so next char will replace
        }
        *to = '\0';//replace the NULL terminator
        strcpy(str, dup);
        free(dup);
    }
    return str; 
}

Example use case:

int main(void)
{
    char line[128] = {"start"};

    while(strstr(line, "quit") == NULL)
    {
        printf("Enter string (\"quit\" to leave) and hit <ENTER>:");
        fgets(line, 128, stdin);
        sprintf(line, "%s\n", strip_ch(line, '"')); 
        printf("%s", line);
    }
    return 0;   
}