What is the Biblical basis for using the names "Yeshua" and "Yehoshua" (Joshua) for Jesus? Messianic Jews commonly refer to Jesus as "Yeshua" or "Yehoshua," which are forms of the Hebrew (and Aramaic) name commonly translated into English as "Joshua."

What is the Biblical basis for using these names for Jesus?
This is, as noted, primarily a question about Semitic languages in the First Century. However, as it happens, there is a Biblical record of the relationship between the Semitic name yĕhôšūaʿ and the Greek Iēsous, the diachronic shift that produced yēšûaʿ, and the equation of yēšûaʿ with Iēsous.1 

Background 

The New Testament is in Koine Greek. Jesus's name here is written as Ἰησοῦς (Iēsous).1 However, it is generally accepted that Jesus spoke primarily Aramaic and/or Hebrew. Determining the precise sounds that Jesus and his friends used to pronounce his name (which is, I suppose, the basis of the preference of the group described in the question) entails speculation about the Hebrew or Aramaic name that corresponds to the Greek name recorded in the Gospels. In this case, whether they were speaking Hebrew or Aramaic is probably not so important. (See below.)

Insights from Septuagint Joshua  

As noted in this question and another related question, the name yĕhôšūaʿ is used most frequently in the Bible to the man who served as Moses's "assistant" during the wilderness period and subsequently led the Israelite people into Canaan, as recorded in the eponymous book. This name is transliterated consistently into the Greek of the Septuagint as Ἰησοῦς (Iēsous). Yĕhôšūaʿ, though, has an additional syllable compared to yēšûaʿ, so we remain one step away from a full explanation.

Insights from Late Biblical Hebrew  

The Biblical books of Ezra, Nehemiah, and Chronicles provide the link we need. These books were composed during the Persian period (539 - 332 BCE) and so occasionally provide information about the transition between the language of the earlier portions of the Hebrew Bible and the dialect spoken in the first century. In these books, the name yĕhôšūaʿ is used only once, in reference to Moses's assistant (1 Chr 7:27). Otherwise, the name yēšûaʿ is used, most often (only?) to refer to the son of Jozadak, a man referred to in Haggai and Zechariah as yĕhôšūaʿ. This occurs both in Hebrew and in the Aramaic portion of Ezra.2 

Starting with Ezra 2:2: 


  ESV: They came with Zerubbabel, Jeshua, Nehemiah...
  Hebrew: ʾăšer-bāʾû ʿim-zĕrubbāḇel yēšûaʿ nĕḥemyāh...
  Greek: hoi ēlthon meta Zorobabel; Iēsous, Neemias...  


Yēšûaʿ, then, is a shortened form of yĕhôšūaʿ that was more commonly used later periods. In addition to being capable of an identical referent, both are translated identically into Greek: Iēsous.3 



Notes  


1. I use throughout the technical transliteration style set out by the Society for Biblical Literature. The transliterations in "normal" Latin characters offered in this question and others are no less "correct" (š = sh), but this system has the advantage of precisely representing each Hebrew consonant and vowel with 1:1 correspondence. 
  


2. In Hebrew: Ezra 2:2, 6, 36, 40; 3:2, 8-9; 4:3; 8:33; 10:18; Neh 3:19; 7:7, 11, 39, 43; 8:7, 17; 9:4-5; 10:10; 12:1, 7-8, 10, 24, 26; 1 Chr 24:11; 2 Chr 31:15. In Aramaic: Ezra 5:2.
     


3. As it happens, in the Septuagint the Greek translation of yēšûaʿ (Iēsous) behaves slightly more like the New Testament's Iēsous than does the Greek translation of yĕhôšūaʿ. Although identical in the nominative case, the Greek term is only partly declinable in the NT: the dative and genitive share the form Iēsou. This is also true in the Greek translations of yēšûaʿ in Chronicles, etc. On the other hand, the dative is (usually) distinguished in the book of Joshua (yĕhôšūaʿ) as Iēsoî.