Assuming that all variables you are looking to rename starts with "Household.Level.form.", you can use rename_with() and gsub():
df_child_level <- tibble("Household.Level.form.One" = runif(1:10),
"Household.Level.form.Two" = runif(1:10))
df_child_level %>%
rename_with(~ gsub("Household.Level.form.", "", .x, fixed = TRUE))