The central legal question would be whether the minor has the capacity and authority to consent to a search: in the context of search law, the police would have to have a reasonable belief that both are the case. It is not reasonable to believe that an 8 year old can consent to a police search, that is, a child will most likely acquiesce to a request from the authorities to conduct a search. It is as reasonable to believe that a 16 year old can give actual consent as it is to believe that an 18 year old can consent. The law leaves the matter open for those under 18, to be determined by circumstances. The other consideration, applicable to younger children, is whether the child is authorized to open the house to outsiders.
So in People v. Hoxter, 75 Cal. App. 4th 406, a 16 year old child invited police in, whereupon polices obtained plain sight evidence of drug offenses by the child's father. The court found that "sufficient discretion certainly exists" by that age. There are similar results in Saavedra v. State, 622 So. 2d 952 involving a 15 year old. In contrast, in Davis v. State, 422 S.E.2d 546, a 10 year old child who was home alone called the police to report drugs in the house. The search was invalidated because
although the child's mother had given him permission to call for
emergency assistance if he needed help, the child had no right, absent
an emergency, to invite anyone into the house while he was alone
there, much less into his parents' bedroom
See also People v. Jacobs, 729 P.2d 757 involving an 11 year old, for extensive discussion of the question of "joint control" and authority to permit a search, citing US v. Matlock, 415 U.S. 164
there must be some objective evidence of joint control or access to
the places or items to be searched which would indicate that the
person authorizing the search has the authority to do so. The mutual
use of the property must be such "that it is reasonable to recognize
that any of the co-inhabitants has the right to permit the inspection
in his own right and that the others have assumed the risk that one of
their number might permit the common area to be searched
This article has a number of citations of relevant cases, which suggests a line for police searches somewhere around 13-14 years old. But also see Lenz v. Winburn, 51 F.3d 1540 for a permitted case of consent given by a 9 year old to a guardian ad litem: the court found a 4th Amendment issue and and concluded that the search was reasonable, holding that minors can give third party consent.
Fourth Amendment rights, unlike rights attendant to due process, do
not guarantee a fair and impartial determination of truth; rather,
they protect the interest of the citizen "to be let alone". Thus, the
subject of a Fourth Amendment-violative search need not be aware of
her right to refuse to give knowing and voluntary consent. However,
the circumstances surrounding the consent must demonstrate that it was
voluntarily given, free of duress or coercion.
