My reason for suggesting putting in some kind of landscape feature with
a berm is that a a rise of a couple of inches might be enough to divert
water that runs down from your neighbor. It would be exceedingly easy
(on a normal sized lot) to lay down that much soil and keep it in place
with sod or other plants. You would need to figure out where the water
will go if you divert it.

We had lots of small problems in the lawn at our condo that I fiddled
with. One was that after many years of running edgers along the same
line, there was quite a hump of lawn along the building with a ditch
next to it. In some places, it had exposed sprinkler pipes that would
then get run over by the mower. In some places I just hit it with a
strong stream of water - the grass stayed in place and the dirt went
into the ditch. Wasted some water ) I found concrete doughnuts 6"
down, with no protection of sprinkler heads that hubby had to keep
fixing. Landscape problems sometimes have simple solutions.

We also had a great deal of erosion behind our seawall - not visible in
places because the sod was so healthy it covered the tunnels between the
washouts - kind of dangerous to walk on. That was fixed by putting some
filter fabric behind the weep-holes in the seawall, gravel behind it,
and filled up with soil.

You may need French drains, but that is a major project. If soil is
saturated, from continuous rain, they might not do any good, but I
really don't know.