Can house be rented in the middle of probate?

No.  She must prove that she is his common law wife to inherit anything and only has authority to rent out property which she owns.  If your brother died without a Will, hire a local probate lawyer to file an Application for Determination of Heirship and Issuance of Letters of Administration.  Don't name her as his spouse.  Let her come in and try to prove that she was.  She will need to show that they both intended to be married (not just that she did), that others in the community believed them to be married (not just her pals), and that they held themselves out as married (for example, by fling federal income tax returns as married, filing jointly).  Assuming she cannot do this, the person appointed by the court to settle (administer) your brother's estate can take charge of the property.