How well does a composite fuselage deal with lightning?
Lightning striking modern aircraft with composite structures


  Composite parts that are in lightning-strike prone areas must have appropriate lightning protection. [...] Composite structures are less conductive than metal, causing higher voltages.
  Lightning protection on airplanes may include wire bundle shields, ground straps, composite structure expanded foils, wire mesh, aluminum flame spray coating, embedded metallic wire, metallic picture frames, diverter strips, metallic foil liners, coated glass fabric, and bonded aluminum foil.
  Lightning can also damage composite airplane structures if protection finish is not applied, properly designed, or adequate. This damage is often in the form of burnt paint, damaged fiber, and composite layer removal.
  
  Source: "Lightning Strikes: Protection, Inspection, and Repair", Boeing, 2012


New technology improving the lightning protection of modern aircraft


  Traditional methods to protect composite aircraft from lightning strike damage rely on a conductive layer embedded on or within the surface of the aircraft composite skin. This method is effective at preventing major direct effect damage and minimizes indirect effects to aircraft systems from lightning strike attachment, but provides no additional benefit for the added parasitic weight from the conductive layer. [...]
  A new multi-functional lightning strike protection (LSP) method has been developed to provide aircraft lightning strike protection, damage detection and diagnosis for composite aircraft surfaces.
  The method incorporates a SansEC sensor array on the aircraft exterior surfaces forming a "Smart skin" surface for aircraft lightning zones certified to withstand strikes up to 100kA peak current. SansEC sensors are open-circuit devices comprised of conductive trace spiral patterns sans (without) electrical connections. The SansEC sensor is an electromagnetic resonator having specific resonant parameters (frequency, amplitude, bandwidth &amp; phase) which when electromagnetically coupled with a composite substrate will indicate the electrical impedance of the composite through a change in its resonant response. Any measureable shift in the resonant characteristics can be an indication of damage to the composite caused by a lightning strike or from other means. The SansEC sensor method is intended to diagnose damage for both in-situ health monitoring or ground inspections. [...]
  The SansEC sensor smart skin technology provides lightning strike protection, shielding effectiveness, and opportunity to achieve in-situ damage detection and diagnostics for aerospace composite structures along with other potential beneficial functions not available from the standard LSP method.
  
  Source: Szatkowski, G. N. at al.: "Open Circuit Resonant (SansEC) Sensor Technology for Lightning Mitigation and Damage Detection and Diagnosis for Composite Aircraft Applications", NASA, 2014 - Download Link


Further information can be found on the internet, see for example "Playing with Lightning in the Name of Aircraft Safety", NASA.

See also this similar question.