Possibly a tight/weakened/overactive or all of the above gluteus maximus muscle - a predominant hip extensor especially involved in uphill running.
1. Is there a specific tender spot to touch (be firm) - if so where, how does it compare to the same area on the other leg?
2. At what point during the motion of running on the affected leg does the pain occur - i.e as the heel is striking the floor / as he is pushing off the affected leg / as the affected leg is back in extension
Try the following - assess for range of movement & pain or both in comparison to the unaffected leg;
1. Laying on back - knee to chest one at a time, pull knee up with hands for full effect
2. In standing - feet wide apart, double leg squat as far as possible (to go further remain on tip-toes)
3. As above but now a single leg squat (balancing on one leg) - use a work surface for upper limbs for balance
4. You (the wife!) watch and assess for this one - get your husband to place HIS finger tips over the bony projections on the front of his pelvis - to find them, run your fingers up the middle of your thighs and keep going until you feel a bony prominence. Now...keeping an upright posture (not bending forward), perform a single leg squat again for each leg. You as the observer note if - his finger tips remain horizontally level throughout the squat in comparison to each leg. Also how 'easy' or 'hard' does the task appear for each leg, any difference, does he shake a lot on the affected side?
To rule out a trochanteric bursitis (very rare and very stubborn to treat) - you the wife, run your hand up the side of the affected thigh (husband in standing), again until you reach a bony prominence sticking out to the side - this is the greater trochanter (not the hip joint itself), and should be near to where the pain is. Stay on the outer most point and press around there very firmly - is there pain, tenderness? If so, compare to the other side, any difference? Hopefully no pain especially as you say the pain is behind the (hip) greater trochanter.
For further assurance that it isn't a bursitis, husband stood side on to a door /wall with the affected leg. Now push to the side against the wall with the affected leg as hard as possible, hold for a few seconds, repeat up to 10 times...any pain/reproduction of the symptoms??
Try the above, note down any differences between affected and unaffected leg and get back to me (PM me if you wish).
Hope this helps
Ian