If you're willing to give spatialite a try (sorry for drifting away from your mysql tag) then you can do as follows. When you say "map each set of coordinates to the neighborhood" I'm assuming you mean to add a column to the coordinate's attrib table with the name of the neighborhood it's located in.
Start with a new, empty spatialite db:
Import the shapefiles. The 'srid' entry is the EPSG code (CRS) for each layer
spatialite> .loadshp "coordinates" coords LATIN1 <SRID>;
spatialite> .loadshp "neighborhoods" nhoods LATIN1 <SRID>;
Add a column to the coords table to accept the neighborhood names. (I guess the neighborhoods layer already has a 'name' column).
Also create a spatial index for each
spatialite> ALTER TABLE coords ADD COLUMN nhood_name text;
spatialite> SELECT CreateSpatialIndex('coords','Geometry');
spatialite> SELECT CreateSpatialIndex('nhoods','Geometry');
Now run an UPDATE query to get the neighborhood names into the coords table
spatialite> UPDATE coords SET nhood_name=(SELECT n.neighborhood_name
FROM nhoods AS n
WHERE ST_Within(pts.Geometry, n.Geometry) AND
(SELECT ROWID FROM SpatialIndex
WHERE f_table_name='coords' AND search_frame=n.Geometry));
That should update the nhood_name column in "coords" such that it contains the neighborhood name in which each point is located.