If a house is being sold within 9 months after the owners death, does the heir (son)have to pay capital gain taxes?

Depends, if it has to go through probate. If there is no will or trust, the property will go through probate and probate estate will need to pay any taxes. If there is a will or trust and property goes to heir, that person may have to include any taxes on income once they send you 1099-C and repost it as income. There are circumstances where taxes will not have to be paid; check IRS publication 4681.

If the property has been transferred to the son then the son will have to pay Cap Gains on any increase in value over the Base value. The Base Value will be the value at the time of death. With only 9 months elapsing there is unlikely to be much change in value.

If the property has not been transferred, then the estate is liable for any taxes, not the heirs.