You can use arcpy to randomly select points by some group attribute. In your case fishnet id.
First Intersect Points with fishnet to get a fishnet id attribute on all Points.
Then modify the commented lines and execute this in the Python window with the point layer added to the map:
import arcpy, random
from collections import defaultdict
points = 'points' #Layer name
groupfield = 'fishnet_id' #Some id identifying each fishnet cell
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.iteritems(): # use groups.items() for ArcGIS Pro/py3
oidlist.extend(random.sample(value,sample_size))
sql = """{0} IN({1})""".format(arcpy.AddFieldDelimiters(points,oidfield), ','.join([str(i) for i in oidlist]))
arcpy.SelectLayerByAttribute_management(points, where_clause=sql)