You have to use the String method .toLowerCase() or .toUpperCase() on both the input and the string you are trying to match it with.
Example:
public static void findPatient() {
System.out.print("Enter part of the patient name: ");
String name = sc.nextLine();
System.out.print(myPatientList.showPatients(name));
}
//the other class
ArrayList<String> patientList;
public void showPatients(String name) {
boolean match = false;
for(String matchingname : patientList) {
if (matchingname.toLowerCase().contains(name.toLowerCase())) {
match = true;
}
}
}