mdfind tool : consider in the same time, to perform researching of files, a substring of the filename and the insensitive case for these substrings

Would mdfind -interpret somestring work?

How 'bout a script?

#!/bin/bash

if [ $# -lt 1 ]; then
    echo "$(basename $0) portion-of-filename"
    exit
fi

find / -type f -print 2>/dev/null | gawk -F'/' -v searchFor="$1" '
BEGIN{
    IGNORECASE=1
}
searchFor {
    base=$NF
    if (index(tolower(base), tolower(searchFor))) {
        printf("%s\n",$0)
    }
}
'