When outbound from an IAF on an approach to a procedure turn, may the pilot choose whether to use time or distance?
Using DME only is perfectly acceptable the FAA does not actually specify how exactly to execute the procedure turn but where and at what altitude. This is covered in the AIM: 


  5-4-9 Procedure Turn and Hold−in−lieu of Procedure Turn. A procedure turn is the maneuver prescribed when it is necessary to
  reverse direction to establish the aircraft inbound on an intermediate
  or final approach course. The procedure turn or hold−in−lieu−of−PT is
  a required maneuver when it is depicted on the approach chart, unless
  cleared byATC for a straight−in approach. Additionally, the procedure
  turn or hold−in−lieu−of−PT is not permitted when the symbol “No PT” is
  depicted on the initial segment being used, when a RADAR VECTOR to the
  final approach course is provided, or when conducting a timed approach
  from a holding fix. The altitude prescribed for the procedure turn is a
  minimum altitude until the aircraft is established on the inbound
  course. The maneuver must be completed within the distance specified
  in the profile view. For a hold−in−lieu−of−PT, the holding pattern
  direction must be flown as depicted and the specified leg
  length/timing must not be exceeded.


The guidance is pretty straight forward that you need to be at the prescribed altitude and within the "remain within" distance but the type and execution of the maneuver is largely up to the PIC as they go on to elaborate. 


  On U.S. Government charts, a barbed arrow indicates the maneuvering
  side of the outbound course on which the procedure turn is
  made.Headings are provided for course reversal using the 45 degree type
  procedure turn. However, the point at which the turn may be commenced
  and the type and rate of turn is left to the discretion of the pilot
  (limited by the charted remain within xx NM distance). Some of the
  options are the 45 degree procedure turn, the racetrack pattern, the
  teardrop procedure turn, or the 80 degree, 260 degree course reversal.
  Racetrack entries should be conducted on the maneuvering side where
  the majority of protected airspace resides. If an entry places the
  pilot on the non−maneuvering side of the PT, correction to intercept
  the outbound course ensures remaining within protected airspace. Some
  procedure turns are specified by procedural track.These turns must be
  flown exactly as depicted


The standard 1 minute timing is a convenient way to execute the maneuver for light aircraft since 1 minute at approach speed is usually ~1.5 miles (assuming a 90 Kt maneuver/approach speed) and will often keep you in the protected area. 

Depending on how you read the regulation in reality a pilot may only use distance, although you may use time as an estimate and helper to execute the maneuver.