I am in the inital process of establishing a single-member LLC in NJ.
My question is how I should describe the Business Purpose of my LLC
for filing purpose. I don't think I need to get into detail of the nature of
the business.

The following passage is extracted from NJ-Business registration
packet p22. "Provide a brief description of the business
purpose for the public record. If the business is a domestic forprofit
corporation, you may leave this field blank and thereby rely
on the general purpose clause provided in N.J.S.A. 14A: The purpose
for which this corporation is organized is(are) to engage in
any activity within the purposes for which corporations may be
organized under N.J.S.A. 14A:1-1 et seq. "

You might look at N.J.S.A 14a ... and see if there are any activities
that you wish to rule out completely.  People starting up often think
they know exactly what their company will be doing and proudly want to
define that as the business purpose, but this restricts the possible
activities and is not recommended.

Just a free comment.
Good luck.