I was told by our ENT to use Nasonex (mometasone) on my son during allergy season. Our insurance doesn't cover it. Can I use Nasocort instead?

Probably. There are many brands of nasal steroids available, all of which are indicated for allergic rhinitis and are interchangeable. However, some people do find some preparations more effective than others.
Allergies . Seasonal allergies can be treated with OTC medications such as generic Zyrtec, Claritin, or Seldane, with generic Flonase and Cromolyn nasal sprays preceded by a plain Saline spray like Simply Saline or Ocean Spray. Reducing exposure to allergens by removing drapery & carpeting, encasing pillows & mattresses, showering & shampooing after exposure to pollens helps too. 'Prevention Works'!
Nasonex (mometasone) vs Nasacort. Simple answer is most likely, yes, since both are nasal steroids but depends upon son's age. Check out http://www.drugs.com/nasonex.html & http://www.drugs.com/nasacort.html for info on both. Flonase (http://www.drugs.com/flonase.html) is another nasal steroid that is available OTC w/o prescription. Better yet, ask your insurer which brand they cover, then ask your ENT to write/change prescription.