The best replacement for haber is tener.

The best alternatives to haber are rated as following:
★★★★★  tener
★★★★  ver
★★★  estar
★★★  hacer
★★  ser
★  ser imperfect
★  suozzi
★  haver
  tom suozzi
  ser and estar
  va a ver

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Speaking about haber, one probably shouldn't forget about past participles.

Other search terms related to haber and their ratings are:
★★★★★ past participles
★★★ past participle in spanish
★★★ past participle
★ impersonal
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Google Trends

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haber
tener
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