What is needed to operate a LSA aircraft at night?
Since you have your private, the LSA just has to be approved.  

According to the FAA Order 8130.2G CHG 1 which covers LSA certification:


  (5) This aircraft is to be operated under VFR, day only, unless
  appropriately equipped for night and/or instrument flight in
  accordance with 14 CFR § 91.205, and when allowed by the
  manufacturer’s operating instructions.


So it must be properly equipped (see below) AND allowed by the manufacturer’s operating instructions.   You will need to check this with each manufacturer to see if they certified it for night flight.

91.205 says day VFR instruments (paragraph b) plus: 


  (c) Visual flight rules (night). For VFR flight at night, the
  following instruments and equipment are required:
  
  (1) Instruments and equipment specified in paragraph (b) of this
  section.
  
  (2) Approved position lights.
  
  (3) An approved aviation red or aviation white anticollision light
  system on all U.S.-registered civil aircraft. Anticollision light
  systems initially installed after August 11, 1971, on aircraft for
  which a type certificate was issued or applied for before August 11,
  1971, must at least meet the anticollision light standards of part 23,
  25, 27, or 29 of this chapter, as applicable, that were in effect on
  August 10, 1971, except that the color may be either aviation red or
  aviation white. In the event of failure of any light of the
  anticollision light system, operations with the aircraft may be
  continued to a stop where repairs or replacement can be made.
  
  (4) If the aircraft is operated for hire, one electric landing light.
  
  (5) An adequate source of electrical energy for all installed
  electrical and radio equipment.
  
  (6) One spare set of fuses, or three spare fuses of each kind
  required, that are accessible to the pilot in flight.