We don't know his name, but I would be willing to bet it was also Henry. Harry is a nickname for Henry (Prince Harry's real name is Henry, for example). The common practice was to name sons after fathers, but also to name boys especially after someone who might pass an inheritance to them.

Why wouldn't Fanny and John have called their own son John? Instead they named him after his grandfather, and I would be very surprised if he (Henry Dashwood) wasn't named after his uncle.

Having a little child who was your namesake could really add to the sentimentality, IMO. It puts me in mind of this passage from Emma (CH. 9) Mr. Woodhouse: "Henry is a fine boy, but John is very like his mamma. Henry is the eldest, he was named after me, not after his father. John, the second, is named after his father. Some people are surprized, I believe, that the eldest was not, but Isabella would have him called Henry, which I thought very pretty of her. And he is a very clever boy, indeed." I can picture Old Mr.Dashwood having similar thoughts/feelings

But even if he didn't, that little boy was still always going to get all the land etc., no matter how much the old uncle cared about his nieces.