Their argument would look like this:
Jesus is working on the sabbath, which only God can work on, being God.
Jesus claims that he is allowed to work on the sabbath because God works on the sabbath, and he works therefore as well, since He is His Son.
Therefore, the sense in which Jesus means He is the Son of God goes beyond the generic 'child of God' sense because it obviously doesn't excuse anyone from breaking the sabbath.
Specifically, the only other sense that makes Jesus' words an argument, is that He is equal with God because He is Son, and vice versa.
The reason that 4 logically follows is that they derive that He is claiming to be equal because He claims to be Son as the reason that whatever God does should apply to Him also. This is how they know He means something unique when He, versus your average Jew, when He says He is "the Son of God." Indeed, this is why the terms "The" Son and "The Father" are used. They clearly refer to a divine Father and Son, and not merely The Divine Father and a creature son. It's "The" Son, and "The" Father. Just as "The" Spirit.
The only way Jesus' argument works is if He is claiming to be, in a word, part of God Himself. As Jesus Himself puts it, they are "one" God. Not two Gods. Not, as Philo the Jew in the first century wrote of the Word of God, "a second God" (although I think we should be charitable with the imprecision of language, and take it in the sense in which it was meant).
We have to first understand Jesus' argument, and John's narration of the event, in order to understand what the Jews meant. Failing to do so is by definition a misinterpretation.
And therefore the Jews persecuted Jesus, and sought to kill him, because he did these things [healing] on a sabbath. But Jesus answered them: My Father works until now, and I too am working. Therefore they sought to kill him all the more, because he not only broke the sabbath, but also called God his own Father, making himself equal with God.
Jesus argues that if God, who created the sabbath, is allowed to work during the Sabbath, then so is the Son of God. The argument is simple, and the Jews pick up on the fact that by "Son of God" (John 10:36) and "my Father" Jesus does not merely mean son of God in a generic sense, but in the fullest sense—that by having God for His Father, He is begotten from God Himself, i.e., "I came forth from God" (John 16:28), and therefore the Jews take issue with the belief that Jesus is the "only-begotten of the Father" "come into the world" (ibid. 16:28) to "make His dwelling among us" (John 1:14). They find it blasphemous because they do not believe that the Son is "God with us" (Matthew 1:23). They do not believe that this Son is "God... become flesh" (John 1:1,14).
They take Jesus' identification as "the Son of God" for a claim to be equal with God because this is what Jesus' argument depends on being true. The logic of His argument is extremely simple: just as God never ceased working during all sabbaths ("works until now"), so is the Son "working." The obvious takeaway is that Jesus is claiming to be the sustainer of all creation just as much as the Father. As indeed the New Testament affirms clearly:"... his Son ... [is] the brightness of his glory, and the figure of his essence, and uphold[s] all things by the word of his power" (Hebrews 1:2-3). "...the Son ... is the image of the invisible God, the firstborn of every creature: for in him were all things created in heaven and on earth, visible and invisible, whether thrones, or dominations, or principalities, or powers: all things were created by him and in him. And he is before all, and in him all things consist" (Colossians 1:13-18). Cf. John 15:1-8.
John 10:27-33 My sheep hear my voice: and I know them, and they follow me. 28 And I give them life everlasting; and they shall not perish for ever, and no man shall pluck them out of my hand. 29 That which my Father hath given me, is greater than all: and no one can snatch them out of the hand of my Father. 30 I and the Father are one.
The argument is again very clear, the Father and the Son are so called because they have "one" nature. Jesus is not some glorified angel, He gives eternal life, and like Jehovah, no one can snatch His elect from His hand. This language is directly from the Old Testament.
Isaiah 43:13 And from the beginning I am the same, and there is none that can deliver out of my hand: I will work, and who shall turn it away?
As elsewhere, Jesus deliberately and clearly conflates the divine work and abilities and prerogatives of the Father to Himself, as can be seen here with the "no one can deliver out of my hand" and "no one can delivered out of My Father's hand." As well as appropriating the image of Israel's Shepherd to Himself, and the prerogative of giving eternal life. This type of conflation is not only obvious, but explicit:
John 5:21 For as the Father raiseth up the dead, and giveth life: so the Son also giveth life to whom he will.
It's no wonder then that, as in the other instance:
The Jews then took up stones to stone him. Jesus answered them: Many good works I have shewed you from my Father; for which of these works do you stone me? The Jews answered him: For a good work we stone thee not, but for blasphemy; and because that thou, being a man, maketh thyself God.
Indeed Jesus does declare Himself one with God, and "the" Son of "the" Father, and thus God by nature, but they are simply wrong to condemn Him—for He was God and man, as the very first words fo the Gospel declare in clear, almost childlike language.
But lastly, note that John's narrative is from the perspective of the writer (John), not the Jews. The sought to stone Jesus all the more because x, y z. Not because they claimed he did x, y or z — Jesus did in fact do every single one of these things.
