Function to find a substring within a string
Sorry but your code is an accident waiting to happen.


Using a constant to declare the available space in you char arrays allows your code to be more maintainable.
scanf() can overflow your buffer and corrupt everything after your buffer.  Forcibly set the BufferSize-1 character to NULL after you have run scanf() its the best you can do with the limitations you have be given.
Why are you declaring index so far up the code, always declare a variable as near to the point of first use as you can, also it could be const, which will make it clear to the reader that its value will never change.
Unless you compiler won't allow it declare loop variables inside the for statement i.e. for (int i = 0; i &lt; momLength; ++i).  The ++i makes no difference for simple types, but when you move on to iterators it does.
Always try and use meaningful variable names, i and j are not very meaningful.  You might read this again in 12 months, so make it easy for yourself.
Now the bad news, I think you algorithm is wrong.  Why are you creating a copy of a string (test)?  I think it would be easier to scan through mum until you find the first character of babe (index).  Then match every character of babe to the characters of mom.  If you get to the end of babe then you have a match starting at index, if a character doesn't match then start scanning mum again at index + 1.
You need to check for zero length mom and babe values entered by the user.
stringLength() could go on "infinitely"  As it stands if you scan a value which overflows the buffer then you could keep going until you run out of memory.  You need to impose a maximum length of BufferSize.
And finally, stick some comments in, it will make it easier when you are getting marked and you'll get a better score.