You can use strstr to find both quotation marks, then use strncpy to copy the correct part of the IP address into the new char array:
char inString[] = "+CIFSR:STAIP,\"192.168.1.117\"";
char IP[16];
char* firstQuote;
char* secondQuote;
Serial.begin(115200);
delay(100);
Serial.println();
Serial.println(inString);
firstQuote = strstr(inString,"\"");
secondQuote = strstr(firstQuote+1,"\"");
char lengthOfIP = secondQuote - firstQuote - 1;
strncpy(IP,firstQuote+1,lengthOfIP);
IP[lengthOfIP] = '\0';
Serial.println(IP);
but this ideally needs some checks to make sure that both quote marks actually exist and that they are a sensible space apart.
[edit]
Just realised there is a 'quicker' function, strtok() http://www.cplusplus.com/reference/cstring/strtok/
which you can use, but my example may help you understand how to do this type of thing more generally.