The best replacement for honours is honors.

The best alternatives to honours are rated as following:
★★★★★  honors
★★★★  masters
★★  masters degree
★★  bachelor degree
★  postgraduate diploma
★  honours
★  ordinary degree
★  bachelors
★  honors canada
★  distinction which is higher
★  distinction
  graduate diploma
  pass degree
  bachelor
  double degree
  double major
  bachelors degree
  basic degree

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Speaking about honours, one probably shouldn't forget about distinction.

Other search terms related to honours and their ratings are:
★★★★★ distinction
★★★ merit
★★ degree
bachelor degree
high honours
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honours
honors
masters