This answer is based on the Bible interpretations written by the Christian theologian Emanuel Swedenborg (1688-1772), which are accepted in the "New Church" or Swedenborgian denominations.
Like other Christian commentators, Swedenborg bases his interpretation of hyssop in the Bible on its use in ceremonial cleansing rituals. However, he interprets it somewhat more abstractly than most traditional Christian interpreters, stating that it refers especially to the cleansing properties of truth in relation to the human spirit.
In Arcana Coelestia ("Secrets of Heaven") #7918 he takes up the meaning of hyssop in the Old Testament as he explains Exodus 12:22, in which hyssop is used to put blood on the lintels and doorposts of the Israelites' dwellings in preparation for the event of the passover. Here is Swedenborg's explanation, in a somewhat archaic translation:
"And you shall take a bunch of hyssop" means the outward means by which purification is effected. This is clear from the meaning of "hyssop" as outward truth, the means by which purification is effected, dealt with below. It says that they were to take a bunch of hyssop because "a bunch" is used to refer to truths and the arrangement of them, (see #5530, 5881, 7408). The reason why "hyssop" means outward truth, the means of purification, is that all spiritual purification is effected by means of truths. For the earthly and worldly types of love from which a person has to be purified are not recognized except by means of truths. When these truths are instilled by the Lord a horror of those types of love, as of things unclean and damnable, is also at the same time instilled. The effect this horror has is that when something similar enters the person's thinking the feeling of horror returns, producing a loathing of those types of love. That is how a person is purified by truths, which serve as outward means.
He then goes on to quote and briefly explain several other Old Testament passages in which hyssop is used in cleansing ceremonies, including the well-known passage in Psalm 51:7:
You will purge me with hyssop and I shall become clean; you will wash me and I shall be made whiter than snow.
To put his explanation in more contemporary terms, also bringing in some of his other explanations of "hyssop":
Hyssop, Swedenborg says, represents the cleansing of our outward self, meaning our wrong actions, and especially the worldly, sense- and self-oriented desires and thoughts that those actions come from. Examples would be the desire for various types of self-indulgence such as overeating just for the pleasure of eating, taking pleasure in casual sex, or a general focus on glamour and excitement—not to mention our desires for specifically prohibited actions such as theft, lying, adultery, murder, and so on that we engage in for our own pleasure and benefit.
These, Swedenborg says, are cleansed by "external truth," meaning the basic commandments of the Bible prohibiting such self-indulgence and prohibiting the committing of these evil and sinful actions. When we accept the truth of those Biblical teachings, this causes us to see the evil thoughts, feelings, and actions we have indulged in as wrong, sinful, and horrible, so that we gain a sense of loathing for everything related to them. In this way, both our earth-oriented desires and thoughts and our outward actions are cleansed from these wrong and evil things.
In Apocalypse Explained #386:30, Swedenborg specifically quotes and explains John 19:28-30, which reads:
Jesus, knowing that all things were now accomplished, that the scripture might be fulfilled, said, "I thirst." And there had been placed a vessel full of vinegar; and they filled a sponge, and put it upon hyssop, and put it to his mouth. And when Jesus had received the vinegar he said, "It is finished."
Here is Swedenborg's exegesis of this passage, once again in a rather archaic translation:
Those who think of these things only naturally and not spiritually, may suppose that they involve nothing more than that the Lord thirsted, and that then vinegar was given to Him; but because all things which the Scripture said concerning Him were then consummated, and He came into the world to save mankind, therefore He said, "I thirst," by which is meant that from Divine love He willed and desired the salvation of the human race. But that vinegar was given to Him, signifies that in the coming church there would be no genuine truth, but truth mingled with falsities, such as exist with those who separate faith from charity, or truth from good, this being signified by vinegar. Their putting it upon hyssop signified a certain purification of it, for by hyssop is signified an external means of purification (as may be seen in Arcana Coelestia #7918).
Here Swedenborg first argues that since this event is presented as fulfilling Scripture, it must have greater meaning that Jesus simply being thirsty and being given vinegar to drink using a sponge and a branch of hyssop.
The scene, Swedenborg says, relates to Jesus' "thirst" to save the human race, which was his purpose in coming into the world.
The "vinegar" relates to the mixture of true and false beliefs that exists within the church on earth. This meaning of "vinegar" is based on "wine," as used in the Holy Supper, meaning divine truth, which is the "blood" of Jesus. Vinegar is a soured form of wine, which causes it to take on the meaning of truth mixed with falsity.
And, expanding a bit on Swedenborg's interpretation here based on statements he makes elsewhere in his writings, the "hyssop" in this passage represents a cleansing of that mixed truth from falsity when it is used in service of the Lord, and especially when it is used in satisfying Jesus Christ's "thirst" to save the human race.
Putting this all together, in our own lives offering Jesus vinegar extended on a branch of hyssop represents our willingness to work in service of Jesus' will and command to reach out with the Gospel to our fellow human beings in an effort to bring them eternal salvation. Though our human understanding of the Gospel may be mere "vinegar," meaning a mixture of true and false understandings of what the Bible teaches about salvation, when we use even that mixed understanding in an effort to satisfy the Lord's desire to save all people, it becomes purified because we are using it from a good heart to do the Lord's work here on earth.
From the perspective of Swedenborg's theology, this purification of our faulty understanding of the Gospel message due to our desire to serve the Lord and do his will in helping to save our fellow human beings is the specific meaning of "hyssop" in John 19:29.