Note that replacing "well-ordered" by "linearly-ordered" produces an equivalent property since any linear order contains a cofinal well order.
Such spaces were called *lob-spaces* and studied by S.W. Davis in [*Spaces with linearly ordered local bases*][1], Topology proceedings 3, (1978), pp.37-51.


  [1]: http://topology.nipissingu.ca/tp/reprints/v03/tp03102.pdf