Not quite-infinitely expandable human-friendly naming system?
As David points out in a comment, you are asking for an impossible thing because the lexiocgraphic order does not have infinite descending chains, and so you will run out of names if you keep adding points on the far left side. So forget the lexicographic order. (You may keep it if you have an upfront upper bound on the number of points.)

You can use the naming scheme x_N_M where N and M are numbers. Think of this as indexing a point with the fraction N/M. Because fractions are dense, you will never run out of them. To insert a new point on the left of the leftmost point x_N_M, name the new point x_N_2M. To insert on the right of the rightmost point x_U_V name the new point x_2U_V. And to insert between adjacent points x_A_B and x_C_D, name the new point x_E_F where E/F is the average of A/B and C/D.