The "household idols/gods/images" are known in Hebrew as teraphim.
In the topic "Teraphim" in the Insight on the Scriptures, gives us a more detailed explanation:
The findings of archaeologists in Mesopotamia and adjacent areas indicate that the possession of the teraphim images had a bearing on who would receive the family inheritance. According to one tablet found at Nuzi, the possession of the household gods could under certain circumstances entitle a son-in-law to appear in court and claim the estate of his deceased father-in-law. (Ancient Near Eastern Texts, edited by J. Pritchard, 1974, pp. 219, 220, and ftn 51) Perhaps Rachel, with this in mind, reasoned that she was justified in taking the teraphim because of her father’s deceptive dealings with her husband Jacob. (Compare Ge 31:14-16.) The importance of the teraphim with respect to inheritance rights would also explain why Laban was so anxious to recover them, even to the point of taking his brothers with him and pursuing Jacob for a distance of seven days’ journey. (Ge 31:19-30) Of course, what Rachel had done was completely unknown to Jacob (Ge 31:32), and there is no indication that he ever attempted to use the teraphim to gain the inheritance from Laban’s sons. Jacob had nothing to do with idols. At the latest, the teraphim would have been disposed of when Jacob hid all the foreign gods turned over to him by his household under the big tree that was close to Shechem.—Ge 35:1-4.
I was able to find an online version of "Ancient Near Eastern Texts Relating to the Old Testament" that the Insight book references and have provided the relevant sections and footnote:
(1) Sale-Adoption47
The tablet of adoption belonging to Kuzu, the son of Karmishe: he adopted Tehip-tilla, the son of Puhishenni. As his share48 (of the estate) Kuzu gave Tehiptilla 40 imers49 of land in the district of Iphushshi. If the land should have a claimant, Kuzu shall clear (it) and give (it) back to Tehip-tilla. Tehip-tilla in turn gave I mina of silver to Kuzu as his honorarium. Whoever defaults shall pay 2 minas of silver (and) 2 minas of gold.
(The names of fourteen persons and the scribe as witnesses, each preceded by the witness-sign.)
(The names of two of the witnesses, one other person, and the scribe, each preceded by "The seal of.")
(2) Sale-Adoption50
The tablet of adoption belonging to Nashwi, the son of Ar-shenni: he adopted Wullu, the son of Puhi-shenni. As long as Nashwi is alive, Wullu shall provide food and clothing; when Nashwi dies, Wullu shall become the heir. If Nashwi has a son of his own, he shall divide (the estate) equally with Wullu, but the son of Nashwi shall take the gods of Nashwi. However, if Nashwi does not have a son of his own, then Wullu shall take the gods of Nashwi.51 Furthermore, he gave his daughter Nuhuya in marriage to Wullu, and if Wullu takes another wife he shall forfeit the lands and buildings of Nashwi. Whoever defaults shall make compensation with z mina of silver and z mina of gold.
(The names of five persons and the scribe as witnesses, each preceded by the witness-sign.)
(The names of four of the witnesses and the scribe, each preceded by "The seal of.")
51 Possession of the household gods marked a person as the legitimate heir, which explains Laban's anxiety in Gen. 31:26 f. to recover his household gods from Jacob. It is to be noted too that Laban binds Jacob in verse 50 to marry no other wives besides his daughters, just as Wullu is bound in our text.
The Jewish Virtual Library, under "Teraphim", has a similar explanation:
The tablets from Nuzi proved to have direct bearing on knowledge of teraphim since the Akkadian term ilāni, "gods," was used in Nuzi legal texts in ways that closely paralleled some of the occurrences of the word ʾelohim or its interchangeable partner teraphim (Gen. 31:30; cf. 31:19, 34, 35). In an adoption contract from Nuzi it is stated that on the death of the adoptive father the adopted son shall be heir. If, however, a natural son is born, he shall be the primary heir and receive his father's ilāni ("gods"); otherwise, the ilāni go to the adopted son. In cases where a normal heir was lacking, the possessor of the ilāni was entitled to a large share of the inheritance.
Rachel's theft of her father's teraphim may be viewed as an attempt to secure her own right to her father's inheritance. Then again, since Laban had begotten sons, Jacob, who may have been adopted by Laban, would have had no right to the gods, and thus Rachel might have stolen them in order to secure the right of paterfamilias for her husband. The idea that possession of the household gods was in some way connected with rights to property inheritance has found widespread acceptance. M. Greenberg, however, has cast serious doubts on the validity of this interpretation, and maintains that since both the adopted son and the legitimate heir divide the inheritance equally, the possession of these household gods does not determine a title to inheritance but rather leadership of the family, and a claim to paterfamilias.
So the "household idols/gods/images" could have been a sort of 'deed' to the family property and/or inheritance.
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