The phase "in the bosom" (κολπον /kolpos) in this context conveys the eternal intimate communion between the Father and Son. Most bibles, even some paraphrases, do not alter the word "bosom". Probably because our English vernacular still uses the word to express the seat of deep affection.
Albert Barnes commentary notes on this verse reads:
In the bosom of the Father - This expression is taken from the custom
among the Orientals of reclining at their meals. See the notes at
Mat_23:6. It denotes intimacy, friendship, affection. Here it means
that Jesus had a knowledge of God such as one friend has of another -
knowledge of his character, designs, and nature which no other one
possesses, and which renders him, therefore, qualified above all
others to make him known. (Albert Barnes Commentary)
There are a few OT references to this phase. When Nathan confronted King David about his adultery, he told him a parable about a beloved ewe lamb.
2Sa 12:3 but the poor man had nothing, save one little ewe lamb,
which he had bought and nourished up: and it grew up together with
him, and with his children; it did eat of his own morsel, and drank of
his own cup, and lay in his bosom, and was unto him as a daughter.(ASV)
In the NT the Apostle John is referred to in the NT as the "one that Jesus loved".
Jo 13:23 There was at the table reclining in Jesus' bosom one of his
disciples, whom Jesus loved.
Jesus no doubt loved all of His disciples, but there was an intimacy between the inner three (Peter, James, John), and in this case, specifically, John, that Jesus did not share with everyone else. He set them apart to show them and teach them things no one else experienced. (For examples see transfiguration and Jesus' selection of the three to pray at Gethsemane).
How is this all related to the Father and Son?
John 1:18 claims that the man Jesus of Nazareth is the unique Son of God. And because of their eternal unity of love, Jesus can truly and fully represent God in every way possible. This fact rings loudly in several of Christ's statements. Such as:
Jo 14:9 Jesus saith unto him, Have I been so long time with you, and
dost thou not know me, Philip? he that hath seen me hath seen the
Father; how sayest thou, Show us the Father?
Or in relation to the Father making all things known to the Son. Just as Jesus made known mysteries to His disciples, and continues to show things to those He loves.
Jo 5:20 For the Father loveth the Son, and showeth him all things
that himself doeth: and greater works than these will he show him,
that ye may marvel.
If you recall, in the scriptures, the only times the Father's voice is heard from heaven is when He is expressing the love of His Son. When the inspired writer wants to convey the idea that loves stems from the deepest part of their being, bosom closeness paints the perfect picture.