Hi
@Patricia C. Here a a few points you might make in response to the verses your cousin has presented:
Deut. 17:17 ( He is speaking to laws of kings) so it may not apply? The same Hebrew word for "multiply" is used in Deut. 17:16 regarding horses. Owning more than one horse was "normal" until the invention of cars, trucks, and tractors. The command does not restrict a king (or anyone else) from owning more than one horse, or one piece of silver or gold, or one woman but rather it restricts kings from hoarding these.
Gen. 2:18 a helper( not multiple). Yes, God made one man and one woman but that doesn't set any limitations on how many men a woman may have nor how many women a man may have; we know the limits through the God's Law. The same reasoning could be applied to the references to a "son" singular in e.g. Proverbs 1:8, 10; 2:1; 3:1, 11, 21, etc.. But just referring to a "son" doesn't set limitations on the number of sons a man may have. God told Noah to take one pair of each kind of animal onto the Ark (Gen. 6:19) but that didn't set a limit for how many pairs of animals there should be nor how many a man may breed. In e.g. Jer. 31:31-32 God refers to Himself as being a husband to more than one wife. Is God presenting Himself as an example of someone doing wrong or being in sin there?
Gen. 2:24 he is to hold fast to his wife and become one! The man is joined to his wife and they become one flesh however there is nothing in the verse to indicate a man may not take another woman and be joined to his (new) wife and they also become one flesh. To read such things into the passage is to create an illusion; it's eisegesis and it builds a doctrine Scripture doesn't support. For example, see also Gen. 2:15; the man was put in the garden to tend and keep but it doesn't mean all men are to be horticulturalists nor does Gen. 2:25; they were both naked.... require all Christians to become nudists. Jesus wasn't married nor was He a horticulturalist or nudist so the Creation account cannot be setting any sort of ideal otherwise Jesus failed to live up to it.
1 Cor 7:1-40 he only speaks of one wife. See my answer for Ge. 2:18 and 2:24 above. Also, you might direct your cousin to 1 Cor. 7:1 where we are told Paul is addressing the question of celibacy, not polygamy! Context, context, context; the three most important rules for understanding the Bible. However, look at v:2, to avoid sexual immorality a man is to have sexual relations with his wife and a woman is to have sexual relations with her husband. Nothing in this passage is dealing with the issue of the number of wives a man may have but it's about avoiding sexual sin.
Eph 5:33 love his wife as himself. Yes, and in Eph. 6:4, fathers are not to provoke their "children" (plu.) but bring "them" (plu.) up. Following your cousin's logic every man who has no child or only one child is in sin. This same response can be applied to the 1 Cor 7:1-40 he only speaks of one wife.
And then the last thing is what does you state law say? Laws to the king did apply to David and Solomon. God NEVER gave the state (human government) authority over marriage, family, or children; never! Just because some organization or religion makes a rule or law doesn't mean you must obey it and especially when that rule or law violates God's Holy Word. The laws that applied to David and Solomon were God's laws not the Egyptian, Philistine, or Canaanite laws.
David had 8 wives and 18 kids he suffered quite a few consequences for this and God didn't allow him to build the Temple because of it! No, David had many wives and concubines, but it was because David had been a man of war and had shed much blood that prevented him from building the Temple (see 1 Chron. 22:8; 28:3). Also, your cousin needs to remember it was through Adam and Eve (a monogamous couple!) that sin came with all its consequences. Tell her to take a close look at her American monogamy-only culture and see the shocking failures of marriages and families there and the consequences of those failures. Check out the name "Hunter Biden" as one to consider.
Hope this is of some help. Blessings and Shalom