The name Lois in the Bible
The name Lois occurs just once in the Bible. In his second letter to Timothy, the apostle Paul reminisces about Timothy's faith, which first dwelled in his grandmother Lois and his mother Eunice (2 Timothy 1:5). Timothy's father was Greek but his mother was Jewish, which probably means that Lois was Jewish as well (Acts 16:1).
Etymology and meaning of the name Lois
It's a bit of a mystery where the name Lois comes from, and it seems to have been invented twice. The modern name Lois may have to do with an ancient Germanic word meaning warrior — although that's doubtful too as Paul's letters to Timothy, and thus the name Lois, has been exceptionally popular for two millennia.
The name Lois shows up first in a Greek text and the only Greek verb that comes close to the name Lois is λωιων (loion), meaning more desirable, more agreeable, and (generally) better (Liddell & Scott, An Intermediate Greek-English Lexicon). It comes from the word λαν (lon), meaning either to seize or hold, or to behold or look upon.
Thus the name Lois would mean Best or Most Beautiful, and to a Greek audience it probably did.
The problem with this explanation is that the name Lois only seems to appear in Paul's letter to Timothy (actually, Paul gives us the dative form Loidi, so readers figure that that means "in Lois"). It's not unusual that a name shows up only once in the entire body of ancient literature, but it has to have come from somewhere. This means that Lois is most likely a name from a language other than Greek, imported and transliterated into Greek. Since Timothy's grandmother Lois was most likely Jewish, the name Lois may very well be a transliteration of a Hebrew name or word or phrase that sounds like it.
The only Hebrew name recorded in the Bible that comes close to the Greek name Lois is ליש (Laish), which is the typically masculine name of a man from Benjamin (1 Samuel 25:44). This name comes from the word ליש (layish), one of a few words meaning lion:
Hence the name Laish means Lion, and lion-names (Leo, Leon, Ari(e)) are not at all unusual in ancient or any times. The name Laish is also a town or region (Judges 18:7, 14) and maybe Lois, or her family, came from there.
The feminine variant of this name, Laishah, occurs in Isaiah 10:30, where it is applied to a town or region. But the objection that a masculine noun can not be applied as a female name is neutralized by the precedent of the name תמר (Tamar — the wife of Er, Genesis 38:6, and the daughter of David and sister of Absalom, 2 Samuel 13), which is identical to the masculine word תמר (tamar), meaning palm tree.
The problem with this approach is that the Septuagint transliterates the name Laish as λαις, spelled with an alpha. Still, even the relation between the words λωιων and its root λαν shows that the aleph sound may not have different all that much from the omega sound. Also, the authors of the New Testament often worked from memory, and it's not uncommon for New Testament quotes of the Septuagint to differ from the original. The names Lois and Laish may even have evolved separately from the original Hebrew word layish.
If Lois was part of a Hebrew speaking community, then her name was probably understood to have meant Lion. But perhaps there was a member of a Hebrew audience who was even more creative, and thought of the Hebrew name אישי (Ishi) which means My Husband (Hosea 2:16). This name comes along in a series of names that also contains לא רחמה (Lo-ruhamah, meaning No Mercy) and לא עמי (Lo-ammi, meaning Not My People), and a creative mind may have invented the name לא איש, Lo-Ish, meaning Without Husband. This invented name may have been readily applied to some old spinster or someone who remained unmarried for religious reasons, but also to someone who had become pregnant without being married. It may be a coincidence but Timothy's maternal grandfather remains unmentioned in Scripture. Perhaps Paul invented the name Lois to make a playful and consolatory point that's not at all unlike his theology. But perhaps not.