The noun σουδαριον (soudarion) describes a handkerchief or sweat-wiping cloth. It's a transliterated Latin term, namely sudarium, which describes a handkerchief for wiping off sweat, that in turn stems from the noun sudor, sweat. This latter noun ultimately stems from the same Proto-Indo-European root "swoyd-" as does our English word "sweat" and the Greek word ιδρως (hidros), see next.

The sudarium was an essential Roman item for people concerned with etiquette, and became a part of Jewish and later Christian attire, namely as a sort of hood or veil worn on the head. Our noun occurs 4 times in the New Testament, see full concordance, twice in relation to the bindings of dead people, which associates our word with the noun οθονιον (othonion), linen.

Also consider that perspiration is essentially the body producing salt water, and that salt was a substance closely associated with the absorption and removal of sickness, disease, sin and ultimately death (see our articles on the nouns αλς, hals and מלח, melah, both meaning salt).

The noun ιδρως (hidros) means sweat and is the Greek equivalent of the Latin word sudor we mention above. It's not formally related but may have reminded a creative audience of the word υδωρ (hudor), meaning water, or even the ever useful term ιδιωτης (idiotes), which means "in a category of their own" (from ιδιος, idios, meaning one's private own or peculiar, as opposed to public or normal).

The latter consideration is possibly even relevant because our noun ιδρως (hidros) occurs in the New Testament in Luke 22:44 only, where we read that Jesus' sweat became like clots of blood dropping to the earth (and see Genesis 4:10 and Matthew 23:35).

Creative commentators have always interpreted this with the image of Jesus squirting blood, but that's because modern English derives likeness from looking alike, whereas classical Greek and Hebrew derive likeness from acting alike. Luke never proposes or implies that Jesus' sweat took on the appearance, color or substance of blood, but rather that his sweat began to behave as if it were his blood. In the Bible, blood (αιμα, haima) is considered the seat of one's soul, and the soul (ψυχη, psuche) is little more than one's breath (see the obvious inverted parallel of Jesus "losing his breath" in the Queen of Sheba losing her "spirit": 1 Kings 10:5). In other words, Luke is telling us that Jesus felt like his soul was dripping away from him (see Matthew 26:38, Judges 16:16, Hebrews 12:4 and of course Isaiah 1:16-18).

Blood is a thing that is supposed to be inside someone, and not outside, and blood on the outside demonstrates a very serious compromise of the living, animated body (and see Ephesians 2:8). Like a person's soul, their blood must be contained within the integrity of the natural body and cannot spill out for any reason. Blood is a person's own hydrological cycle, a mini-version of the hydrological cycle of the whole earth if you will (see our article on the noun νεφελη, nephele, meaning cloud, for more on the mental equivalent of hydrological cycle). One may even say that, likewise, one's own mind is (or is supposed to be) a mini-version of God's universal mind (John 17:21-24; this is called self-similarity). This is why we don't want to have anything on our minds that God doesn't want on his (Revelation 3:16). Jesus sweat dropping like blood to the earth is not about sweat transubstantiating into blood but about purifying the greater Body of Christ (1 Corinthians 12:27).