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The Republic of Texas would have been Texas more official name as indicated by its constitution in 1845 required by all states that wish to join the union.
We, the people of the republic of Texas, acknowledging with gratitude the grace and beneficence of God, in permitting us to make a choice of our form of government, do, in accordance with the provisions of the joint resolution for annexing Texas to the United States, approved March first, one thousand eight hundred and forty-five, ordain and establish this constitution.
Texas was informally known as the lone star republic as a nickname because of its similarities with the U.S goverment and the large number of U.S citizens that lived in Texas and wanted to be annexed by their home country. The Lone Star republic served as a patricotic call to the U.S to annex inhabitants who wanted the protection of the mighty U.S goverment from the Mexicans they had barely secured independence from. This idea is furthered by the fact that Texas changed its flag twice from its revolutionary Burnet Flag to a more U.S similar flag officially known as the Lone Star flag(found here). Finally the name Texas itself comes from the native American tribe Caddo which means friends or allies(explained here) which was adopted by the Spaniards upon first settling the area.
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It was called The Republic of Texas. The Wikipedia article on The Republic of Texas states:
The Republic of Texas (Spanish: RepΓΊblica de Tejas) was a sovereign state in North America that existed from March 2, 1836, to February 19, 1846. It was bordered by Mexico to the west and southwest, the Gulf of Mexico to the southeast, the two U.S. states of Louisiana and Arkansas to the east and northeast, and United States territories encompassing parts of the current U.S. states of Oklahoma, Kansas, Colorado, Wyoming, and New Mexico to the north and west. The citizens of the republic were known as Texians.
The Mexican province of Tejas (in English history books usually referred to as Mexican Texas) declared its independence from Mexico during the Texas Revolution in 1836. The Texas war of independence ended on April 21, 1836, but Mexico refused to recognize the independence of the Republic of Texas, and intermittent conflicts between the two states continued into the 1840s. The United States recognized the Republic of Texas in March 1837 but declined to annex the territory.
From the Wikipedia article on Texas regarding the etymology of Texas:
The name Texas, based on the Caddo word taysha (or tayshas) meaning "friends" or "allies", was applied by the Spanish to the Caddo themselves and to the region of their settlement in East Texas.
It's form of government was a republic, hence the name Republic of Texas.