import arcpy, random
from collections import defaultdict

points = r'points' #Layer name
groupfield = 'fishnet_id' #Some id identifying each fishnet cell spatially joined
sample_size = 1 #Points to select per group

oidfield = arcpy.Describe(points).OIDFieldName

groups = defaultdict(list)

with arcpy.da.SearchCursor(points, [groupfield, oidfield]) as cursor:
    for key,value in cursor:
        groups[key].append(value)

oidlist = []
for key,value in groups.items():
    oidlist.append(random.sample(value,sample_size)[0])

sql = """{0} IN({1})""".format(arcpy.AddFieldDelimiters(points,oidfield), ','.join([str(i) for i in oidlist]))
arcpy.SelectLayerByAttribute_management(points, where_clause=sql)