I've seen the Henry that I discovered here, described as "the Writer" in one or two of the other biographies of this very prominent family. Also,
"It was during the course of the Parliament that [Sir Walter] Rice’s financial problems caught up with him. Outlawed for debt, he was omitted from the Carmarthenshire commission of the peace in September 1607. His financial difficulties may also explain his absence from Westminster in July 1610. On 13 Mar. 1612 Rice passed his Pembrokeshire estates to his eldest son,
Henry, who agreed to pay off his debts, which by now amounted to more than £2,500.32 The list of his creditors included prominent local gentlemen such as Sir Henry Jones of Abermarlais (£400) and his own son-in-law, Thomas Button (£100), as well as several London merchants. It even included the warden of the Fleet (£60), suggesting that Rice had recently been incarcerated for debt. Attempts to remortgage the Pembrokeshire estates seem to have failed, but in 1617 Henry Rice secured a lease of the Crown’s extent, which forced the creditors to an agreement in the following year. Chief among these was Button, who assigned his naval pension to Henry Rice, and agreed to take another of Sir Walter’s sons to sea with him.
In the early 1620s Henry Rice negotiated to marry Mary, a daughter of Sir Thomas Myddleton I*, hoping this alliance would provide £10,000 to help cover his father’s debts.
https://www.historyofparliamentonline.org/volume/1604-1629/member/r...
So the wife of Henry Rice (son of Sir Walter Rice), is known.