You can replace either by inspection or solving for the new variables:

h = (t1 - t2)^2 + (t2 - t3)^2 + (t1 - t3)^2 + (m1*(ob1 - on))^2/
    m1 + (m2*(ob2 - on))^2/m2 + (m3*(ob3 - on))^2/m3;
Simplify[h /.
  Solve[{e1 == t1 - t2, e2 == t2 - t3,
     p1 == m1*(ob1 - on), p2 == m2*(ob2 - on), p3 == m3*(ob3 - on)},
    {t2, t3, ob1, ob2, ob3}][[1]]]
Grad[%, {e1, e2, p1, p2, p3}]

Your three components of eta are not independent.

The clean way is to start with H in terms of eta and p from the beginning. However, you can replace back:

h = (t1 - t2)^2 + (t2 - t3)^2 + (t1 - t3)^2 + (m1*(ob1 - on))^2/
    m1 + (m2*(ob2 - on))^2/m2 + (m3*(ob3 - on))^2/m3;
Simplify[h /. 
  Solve[{e1 == t1 - t2, e2 == t2 - t3, p1 == m1*(ob1 - on), 
     p2 == m2*(ob2 - on), p3 == m3*(ob3 - on)}, {t2, t3, ob1, ob2, 
     ob3}][[1]]]
Grad[%, {e1, e2, p1, p2, p3}] /. {e1 -> t1 - t2, e2 -> t2 - t3,
  p1 -> m1*(ob1 - on), p2 -> m2*(ob2 - on), p3 -> m3*(ob3 - on)}

You have to make up your mind on the three components of eta.