Since you're already using StringUtils, StringUtils.replaceIgnoreCase is a good candidate to go with. Worth to mention that the version 3.5+ required.
public static String replaceIgnoreCase(String text,
                                       String searchString,
                                       String replacement)
Case insensitively replaces all occurrences of a String within another
  String.
A null reference passed to this method is a no-op.
 StringUtils.replaceIgnoreCase(null, *, *)        = null
 StringUtils.replaceIgnoreCase("", *, *)          = ""
 StringUtils.replaceIgnoreCase("any", null, *)    = "any"
 StringUtils.replaceIgnoreCase("any", *, null)    = "any"
 StringUtils.replaceIgnoreCase("any", "", *)      = "any"
 StringUtils.replaceIgnoreCase("aba", "a", null)  = "aba"
 StringUtils.replaceIgnoreCase("abA", "A", "")    = "b"
 StringUtils.replaceIgnoreCase("aba", "A", "z")   = "zbz"
In your case:
String[] old = {"ABHISHEK", "Name"};
String[] nw = {"Abhi", "nick name"};
String s = "My name is Abhishek";
for (int i = 0; i < old.length; i++) {
    s = StringUtils.replaceIgnoreCase(s, old[i], nw[i]);
}
System.out.println(s);
Output:
My nick name is Abhi
You can even create a helper method if you going to use it frequently:
public static String replaceIgnoreCase(final String s, final String searchList[], final String replacementList[]) {
    if (searchList.length != replacementList.length)
        throw new IllegalArgumentException("Search list and replacement list sizes do not match");
    String replaced = s;
    for (int i = 0; i < searchList.length; i++) {
        replaced = StringUtils.replaceIgnoreCase(s, searchList[i], replacementList[i]);
    }
    return replaced;
}
And use it like you would use a library call:
s = replaceIgnoreCase(s, old, nw);