I have to admit to having a bit of a thing for Eu names. It started with Eugenie and Eulalia and went from there.
Many Eu- names come from the Ancient Greek element εὖ meaning "good, well" In English it is transliterated either as "Ev" when followed by a vowel (e.g. Evadne) or as "Eu" when followed by a consonant. There is also ευρυς (eurys) meaning "broad, wide."
This "Eu" diphthong is generally pronounced (in English) with a "yoo" sound.
Eudokia -- "to seem well." It is more commonly known by it's Latinised form Eudocia.
Eudora -- "good gift." The name of one of the Neriads (sea-nymphs) in Greek mythology.
Other Eu- Neriads include Eukrante / Eucrante "successful," Eulimene "good harbour," Eumolpe "fine singing," and Eupompe "fine procession."
Eudoxia -- "good fame." Aelia Eudoxia was a Byzantine empress.
Eugenios -- "well born." Eugene is the common Anglicisation, while Eugen (Germany) and Eugenio (Spanish, Italian) are used elsewhere. The feminine Eugenia / Eugeneia is also found in the forms Eugenie (French) and Eukene (Basque).
Eukleides -- "good glory." Eukleides of Athens was a renowned sculpture, while the paired down form, Euclid, is used for the famous Greek mathematician.
The feminine form Eukleia / Eucleia was the name of the goddess of glory and good repute. Her sisters were Eupheme "praise, acclamation," and Euthenia "prosperity."
Eulalia -- "to talk well." Eulalie is the French form, while Eula, Eulia, Laia, Lalia and Lalla are found as short forms.
Eumelia -- "good song, melody"
Eunike -- "good victory." Another of the Neriads, known better in the Latinised form Eunice.
Eunomia -- "good order, good law." The goddess of good order and lawfulness. She was also associated with green pastures (nomia meaning pasture). Her sisters were Dike "justice" and Eirene "peace."
Euphemia -- "to speak well, acclamation." Effie is a well establish diminutive, but Phemie and Eppy have also been used.
Euphrasia -- "good cheer" is also also the name of a plant genus, also known as Eyebright. Euphrasie is the French form.
Euphrosyne -- "merriment." The appropriately given to the Greek goddess of joy, good cheer and merriment. She was one of the graces alongside Aglaia "beauty" and Thalia "festivity, flourishing."
Eupraxia -- "good action." The spirit of good conduct and good actions.
Euripides -- "good throw." The name of the famous Classical poet.
Europe -- "wide eyed" or "wide face." Latinised as Europa. The name of a beautiful Phoenician woman who was abducted by Zeus in the guise of a cow.
Euryalus -- "wide sea." A name borne by severeal characters in Greek mythology. The feminine Euryale was borne by one of the Gorgons.
Euryanassa -- "wide (reaching) queen." In Greek mythology this was the name of the wide of Tantalus.
Eurybia -- "wide force." A sea goddess as well as a genus of butterfly and plant.
Eurydike -- "wide justice." Latinised as Eurydice and known as the famous, and beloved, wife of Orpheus.
Eurymedon -- "wide (reaching) ruler." The name of an Athenian general in the Peloponnesian War.
Eurynome -- "broad pastures" or "wide ruling" Two figures feature in Greek mythology with this name. The first was an early Titan queen of Olympus, who was later deposed by Rhea and Kronos. The second was the goddess of pastureland, and mother of the Kharites (Three Graces).
Eusebia -- "pious." The spirit of piety, loyalty and duty. She was the wife of Nomos "law." Eusebios is the masculine form, which has been rendered as Eusebio in Spain and Italy.
Eustathios -- "staedy." More commonly known in the Anglicised form Eustace, and the feminine Eustacia. Another possibly source is Eusthycus "fruitful".
Euterpe -- "well pleasing, giving much delight." One of the nine Muses, and goddess of song, music and dance.
Euthalia -- "fine bloom." A 3rd century saint and genus of butterfly.
Euthemia -- "neat, well ordered." A Naiad nymph of Kos.
Euthymios -- "good spirits." Euthymios the Great was an early Byzantine saint. Eutimio is the modern Spanish form.
The feminine Euthymia was used as an alternate name for the goddess Euphrosyne.
Eutropius -- "versatile." Both Eutropius and Eutropia were used by many prominant figures in the Byzantine Empire.
Eutychus -- "fortunate." This name appears in the New Testament as a man who fell out of a window while Paul was preaching but miraculously survived. The name was also borne by an early Christian theologian.
And then there are Welsh Eu- names, which are a whole different kettle of fish all together. These names derive from the Welsh aur meaning "gold" and is pronounced with an "ay", followed by a rolled 'r', sound.
Eurddolen (ayr-THOL-en) -- "gold link"
Eurfron (AYR-vron) -- "gold breast"
Eurfryn (AYR-vrin) -- "gold hill"
Eurgain (AYR-g'eye'n) -- "gold beauty."
Eurig, Eurion -- "gold."
Eurliw --(AYR-leeoo) "gold colour." A unisex name.
Eurof (AYR-ov)-- "gold smith."
Eurolwyn (ayr-OL-win)--"gold wheel." A character mentioned in the tale of Culhwch and Olwen.
Euron (AYR-on)-- "gold." The name of the beloved of legendary poet Iolo Goch. Euronwy was a name given to a mythical enchantress.
Euros (AYR-os) -- "gold." Euros Bowen was a notable Welsh poet. Euroswy and Euroswen are found as feminine forms. Euros also features in Greek mythology as the god of the East Wind.
Eurwen, Eurwyn (AYR-win) -- "gold + white, fair, blessed."
Eurydd (AYR-ith) -- "goldfiner."
Euryl (AYR-il) -- "gold jewel."
Euryn (AYR-in) -- "gold trinket."
The Victorians also coined such gems as Euretta, Eurania and Eura.