🔼The name Jesse in the Bible
Jesse the Bethlehemite is the son of Obed, the son of Boaz and Ruth. Jesse is the father of eight (1 Samuel 16:10-13) or seven (1 Chronicles 2:15) sons, namely Eliab, Abinadab, Shimea, Nethanel, Raddai, Ozem, and David, and two daughters, Zeruiah and Abigail. 2 Samuel 17:25 ascribes these two daughters to Nahash, which may mean that Nahash is another name of Jesse.
Having two names is not uncommon in the Bible. The father of Zipporah, for instance, is called Reuel in Exodus 2:18 and without fanfare Jethro in Exodus 4:18. It may also mean that Nahash and Jesse are different people, and that either Jesse or Nahash is the natural father while the other adopted them. This possibility becomes quite likely when we assume that Nahash was in fact a son of Jesse (number eight; born after David, who was the youngest in 1 Samuel 16), which in turn neatly solves the problem of the number of Jesse's sons.
Saying daughter for grand-daughter or even more remote descendants is not uncommon in the Bible (2 Chronicles 2:14 — 'daughters' of Dan; Luke 1:5 'daughters' of Aaron). Possibly, Nahash died and his daughters were raised as if they were Jesse's. Another possibility is that Nahash wasn't noticed much and that his grandsons, who were all heroes of the army, rather referred their filiation to famous Jesse, the father of the king, than of Nahash Nobody.
If Nahash was indeed Jesse's youngest, he was probably held in the same low esteem as David was. When Samuel came to Jesse with the intention to anoint the next king of Israel, Jesse assumed it would be his first-born. Young David wasn't even called to the house to witness this great event, until it became clear it was he who Samuel was after.
The prophet Isaiah reported that the Messiah would come forth from Jesse (Isaiah 11:1 and 11:10), and since he indeed became an ancestor of Christ, his name also appears in the Greek New Testament (spelled Ιεσσαι, Iessai, it's used 5 times in the New Testament; see full concordance).
🔼Etymology of the name Jesse
The meaning of the name Jesse is rich enough to ignite debate. In 1 Chronicles 2:13, Jesse is called by the similar name אישי, Ishi, meaning My Husband (see Genesis 29:32, Hosea 2:16) and it may be that both the names Ishi and Jesse are formed from the word איש ('ish), meaning man:
It is then noted that the phrase אישי is only one letter beth away from the name Abishai (אבישי), meaning My Father Is Jesse, and thus My Father Is My Husband.
Another way of explaining the name Jesse may be in the combination יה, which is short for יהוה (YHWH; the Name of the Lord), and the enigmatic particle יש (yesh), asserting existence:
Jones' Dictionary of Old Testament Proper Names explains the name Jesse solely through the word יש but seems to derail a bit in the explanation. The word יש can't be literally translated in English because it expresses the existence of the word or phrase that follows it. In Proverbs 8:21 the word occurs so that the literal translation would be something like: the grain houses are full of whatever it is that grain houses get filled with". The latter part is commonly translated with wealth or abundance, but this is implied by the context and not by the word יש.
Jones freely assumes that Jesse was a man of substance and so translates his name with Wealth. This name, however, does not imply this. If it comes from the word יש it stresses Jesse's presence in the world. Maybe the name Jesse is the Hebrew equivalent of pop-appellatives like The Fonz or The Man.
The name Jesse holds a wide pallet of meaning, especially since no reader should feel coerced to choose just one. The name Jesse means My Husband, as well as Yah/Yahweh Exists (NOBSE Study Bible Name List). As such the name Jesse contains the most profound notion that human marriage reflects divine revelation.