What do you mean by the term strict part of a binary relation?
How can it be used to define minimal element for any set with relation?
It's hard to say without more context, but it seems like you are to take out pairs of the form $(x,x)$ from your relation (i.e. dropping the condition of reflexivity). This is similar to how "less than or equal to" gives rise to "strictly less than".
Given a strict relation, you can find a minimal element of a finite set by taking a descending chain $x_1 > x_2 > \cdots > x_n$. Since the set is finite, the chain will indeed terminate at a minimal element.
If the set is infinite, there may not be a minimal element under every relation. For example, in the real numbers with usual "less than" relation there can be descending chains with no minimal element.