If I sell a house to my unwed daughter and she brings her boyfriend to live there who is also the father of my granddaughter, and she charges him rent to live there, can he claim some partial ownership to the house if they split up at a later date?

As it's charged rent then he cannot claim any sort of ownership at all.

If there is a provable lease agreement (oral or written, preferably written) then he cannot claim any ownership. If he is not on the deed, the only way he MIGHT be able to claim ownership is if they marry and he claims it as community property.