From Biblical and historical studies we have found the following information about James the Lesser:

• He was the son of Alpheus
• He lived in Galilee
• Possibly the brother of the Apostle Jude
• He wrote the epistle of James
• Preached in Palestine and Egypt
• He was not as well known as the other apostles
• James the Lesser was only mentioned three times in the bible.

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• Matthew 10:3
• Mark 3:18
• Luke 6:15
• Acts 1:13

Some references believe he could have been the brother of St. Matthew (the tax collector) but this has not been proven. James was considered of strong character with an intense personality which was often referred to as "fiery".
James is the son of Mary 
Mark 15:40 (where he is called ò mikros "the little", not the "less", as in the D.V., nor the "lesser"); Matthew 27:56. 
Possibly identitified with James the Less
"James, son of Alphaeus is often identified with James the Less, who is only mentioned three times in the Bible, each time in connection with his mother. Mark 15:40 refers to "Mary the mother of James the younger and of Joses", while Mark 16:1 and Matthew 27:56 refer to "Mary the mother of James". James, son of Alphaeus may also be identified as James the Just (1). Since there was already another James (James, son of Zebedee) among the twelve apostles, equating James son of Alphaeus with James the Less made sense. (James son of Zebedee was sometimes called "James the Greater").
Many Modern Biblical scholars are divided on whether this identification is correct. John Paul Meier finds it unlikely.
Fresco of Saint James the Less in the Orthodox Church of Vladimir, Russia. 12-th century.]], Eastern Orthodox and Protestants tend to distinguish between the two. 
Possibly the brother of Matthew
Another Alphaeus is also the name of the father of the publican Levi mentioned in Mark 2:14. The publican appears as Matthew in Matthew 9:9, which has led some to conclude that James and Matthew might have been brothers.
 The four times that James son of Alphaeus is mentioned directly in the bible (each time in the list of the Apostles) the only family relationship stated is that his father is Alphaeus.In two lists of the Apostles, James and John are listed as brothers and that their father is Zebedee (James, son of Alphaeus. 2019).
How James the Lesser Died 
Records indicate that he was crucified in Egypt and that he died as a martyr because he was cut into pieces with a saw.
​James Apostolic Symbol
The Saw" (1).

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Apostles of Christ

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Memory Verse

And the wall of the city had twelve foundations, and in them the names of the twelve apostles of the Lamb" Revelations 21:14  

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References:

1. ​James, son of Alphaeus. (2019). En.wikipedia.org. Retrieved 20 January 2019, from https://en.wikipedia.org/wiki/James,_son_of_Alphaeus
2.  ​Christian, A. (2010). Jesus’ Disciples Names: List, Meaning and Occupation. Bibleseo.com. Retrieved 20 January 2019, from http://bibleseo.com/gospels/jesus-disciples-names-meaning-occupation/
3. James, brother of Jesus. (2015). En.wikipedia.org. Retrieved 20 June 2019, from https://en.wikipedia.org/wiki/James,_brother_of_Jesus
4. HOLY BIBLE: Mark 3. (2019). Newadvent.org. Retrieved 20 June 2019, from http://www.newadvent.org/bible/mar003.htm#vrs18