In the Hebrew Bible, why did God pick the Jews as God's chosen people?
This is a question that necessitates a long exposition but given the fact that you have been charged with finding answer to this issue, here are some thoughts you may want to share with your Bible Study group.
The traditional answer is that God's choosing of Israel was an undeserved status conferred upon Israel based on God's grace, not on Israel's merits. There is no logical explanation. Israel was no better than other nations and yet God chose her. We can stop here and say that it is all part of God's mysterious ways. No one can explain it. That is the traditional theological answer. The apostle Paul expresses it very well in Romans 11:33. "O the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!"
Believers of each of the monotheistic religions - Judaism, Christianity, and Islam - claim to have been chosen by God. When one believes that there is only one God, then there has to be only one people that can claim absolute access to that God. The Hebrew Bible attests to this in Deuteronomy 6:4: "Hear, O Israel: The Lord our God is one Lord." The New Testament says something similar: "For there is one God; there is also one mediator between God and humankind, Christ Jesus, himself human." (1 Timothy 2:5). And the Holy Qur'an asserts: "And your God is one God; there is no God but He, the Gracious, the Merciful." (2:164).
Exodus 6:7 provides an early witness to this idea of Israel having been chosen by God: "I will take you as my people, and I will be your God." The rest of the OT confirms this. In the New Testament a similar thought is presented: "But you are a chosen race, a royal priesthood, a holy nation. God's own people." (1 Peter 2:9). And the Qur'an resonates with the same idea: "Allah chooses His Messengers from among angels, and from among men." (22:76)
But rivalries arise when you are faced with these three exclusive claims. Who is right? Who are God's chosen people, the Jews? the Christians? the Islamic people? As you can see, this is a very dangerous thought that has justified many religiously motivated conflicts in the world, from the Crusades, to the Inquisition, to the Conquest of the Americas, to the Holocaust, to the Palestinian/Jewish conflict, to mention just a few. If we interpret the concept of a chosen people as historical, that is, that the Creator God actually took sides and chose a people over another, well, that will make God very limited and prejudiced and will lead us into the problem outlined above. But if we say that that is how an oppressed people experienced God, then it is different. That is, if we interpret it theologically rather than historically, and contextually rather than universally, then we can see how this idea can bring comfort to the lives of troubled men and women.
So, the short answer is: God did not pick the Jews only. God has picked, that is, is willing to work with the faithful of any religion who are willing to invest their lives for the betterment of the human race.
The Talmud suggests that all the other nations were offered the Torah, and only the Jews accepted.
Other interpretations are that Jews are particularly 'Israel' - which means literally those who wrestle with God (the name given to Jacob after his dream/vision of wrestling with some divine being). So God chose a people who will not just worship, but also struggle. Cf. Abraham arguing with God over punishing Sodom and Gemorrah.
Some Jews say - "If we are the chosen people, maybe He can chose someone else for a while."
My brother - Dovid Gottlieb - who is an ultra ultra orthodox rabbi, thinks the long existence of our tiny group shows how chosen by God we are. My reply is that maybe God is keeping us alive to torture . . . and that in any case, using numerical proofs of a divine order is simply a category mistake.
Best for your holidays, however you observe.