import bpy

for i, obj in enumerate(bpy.context.selected_objects, 1):
    bpy.context.scene.objects.active = obj
    obj.name = "object" + str(i)

Just replace "object" with "your name" and run this in the text editor (make sure you have all objects selected that you want renamed).

Blender 2.79

import bpy

for i, obj in enumerate(bpy.context.selected_objects, 1):
    bpy.context.scene.objects.active = obj
    obj.name = "object" + str(i)

Blender 2.8+

import bpy

for i, obj in enumerate(bpy.context.selected_objects, 1):
    bpy.context.view_layer.objects.active = obj
    obj.name = "object" + str(i)

Just replace "object" with "your name" and run this in the text editor (make sure you have all objects selected that you want renamed).