Solution: From "MN=NP=PR= 1 and point A is between M and N, and point B is between P and R", it can be concluded that the arrangement order of points on the number axis is: M < A < N < P < B < R；;

And each point is an integer point, then we can get: b-a = 2 ①.

Let's discuss |a|+|b|=3 ②:

When a≥0 and B > 0, ② becomes a+b=3 ③, which can be obtained as follows: a=0.5, b=2.5, and the origin of the number axis is m;

When a < 0 and b≥0, equation ② becomes: b-a=3, which contradicts equation ① and is untenable;

When a < 0 and b < 0, ② becomes: -a-b=3 ④, and it can be obtained that a=-2.5 and b=-0.5. At this point, the origin of the number axis is R.

To sum up, the origin of the number axis should be point M or point R.