What basis is there for defining sin as "rebellion"? Many popular Protestant gospel explanations and frameworks define the essence of sin as rebellion against God.
Two Ways to Live:

The sad truth is that, from the very beginning, men and women everywhere have rejected God by doing things their own way. We all do this. We don’t like someone telling us what to do or how to live—least of all God—and so we rebel against him in lots of different ways. We ignore him and just get on with our own lives; or we disobey his instructions for living in his world; or we shake our puny fists in his face and tell him to get lost.
How ever we do it, we are all rebels, because we don’t live God’s way. We prefer to follow our own desires, and to run things our own way, without God. This rebellious, self-sufficient attitude is what the Bible calls ‘sin’.

Knowing God Personally:

&quot;All have sinned and fall short of the glory of God&quot; (Romans 3:23). Man was created to have fellowship with God; but, because of his own stubborn self-will, he chose to go his own independent way and fellowship with God was broken. This self-will, characterized by an attitude of active rebellion or passive indifference, is an evidence of what the Bible calls sin.

Christianity Explored:

But many people rejected Jesus. They thought they’d be happier making their own rules and living outside his kingdom. This rejection of the King is something we all do. Jesus called it sin.

But the Hebrew and Greek words for sin have the senses of failure, error, and evil.
What basis is there for defining sin as &quot;rebellion&quot;?

Is there any lexical support for such a definition (either through a sub-sense of those words, or through other words)?
Are there any passages which define or equate sin and rebellion?
I would like to add to the answers already posted a few observations about the theological lexical register of the Biblical writers that may bear on the OP's question.


  Is there any lexical support for [sin as rebellion] (either through a sub-sense of those words, or through other words)?


Yes. This is true both in the OT and the NT. The NT understanding of sin is the development of a Hebrew concept, and the vocabulary follows suit. Thus, although, as pointed out by the OP, the Greek ἁμαρτία (and a slurry of cognates) in Ancient Greek denoted "missing the mark", they are frequently used in the LXX to represent the particular Jewish notion of sin.1  In the NT these words are never used without these moral connotations (NIDTTE). This Hebrew background is therefore of primary importance in understanding the meaning of the NT authors.

In Hebrew, there are indeed other words conveying the notion of sin.  The OP mentions חטא (ḥṭʾ), which can be used outside the moral realm to mean "to be mistaken" or "to miss". (See, for instance, Judg 20:16, of archers who could sling stones and not "miss" (ḥṭʾ).) The term is also used to express interpersonal ethical lapses (e.g. 2 Kings 18:14). The most common use in Biblical Hebrew, however, is in the theological sphere, where it denotes failure in one's obligations toward God, particularly in the legal and cultic domain. This is the most common word used for sin, used ~595 times. It contains no a priori notion of rebellion.

However, there are many other words that express related concepts of human evil sometimes translated "sin".2  The most relevant here is the term פשע (pšʿ) (~145 times). Unlike ḥṭʾ which can refer to a mere mistake (primarily cultic inadvertences), pšʿ essentially means "rebel", also sometimes translated "revolt" or "transgress".3   In interpersonal relationships, it means "willful violations by an inferior against a superior" (Luc; see, e.g., Gen 31:36, Prov 28:24). In man's relationship with God, this sense is preserved:


  In biblical theology, the term refers to an open and brazen defiance of God by humans. 


This, then, is the word that denotes the aspect of sin that comes closest to the notion of rebellion. Not surprisingly, it is a favorite of the (latter) prophets, where the role of Israel's pšʿ in their broken relationship with Yahweh is relentlessly rehearsed. 


  These prophets...make repeated use of pšʿ because their ministries devote significant attention to Israel’s past or present covenant treachery. They indict Israel for disrupting their covenant relationship with Yahweh. Isaiah begins his prophecy by depicting Yahweh’s relationship with Israel in terms of a father-son relationship. Israel, although a recipient of Yahweh’s abundant, tender care, has rebelled against her father (suzerain) (Isa 1:2) and faces severe judgment (1:28) (Carpenter).


A few (of many available) examples that well illustrate this theme:

Hos 7:13:


  Woe to them, for they have strayed (or fled) from me!
  &nbsp;&nbsp; &nbsp; &nbsp; 
          Destruction to them, for they have rebelled (pšʿ) against me!


Hos 8:1:


  because they have transgressed (lit. passed by) my covenant
  &nbsp;&nbsp; &nbsp; &nbsp;
          and rebelled (pšʿ)  against my law. 


Isa 1:2:


  Children have I reared and brought up,
  &nbsp;&nbsp; &nbsp; &nbsp;
          but they have rebelled (pšʿ) against me. 


Rebellion is thus central to pšʿ. 

The OP further asks:  


  Are there any passages which define or equate sin and rebellion?


Other answers have given many relevant conceptual parallels, the connection with Genesis 3 (which uses neither of these terms!4 ) being the strongest in my mind. This is summarized by Bloesch: 


  It should be acknowledged that sin entails both privation and positive rebellion, but the latter is prior to the former. "The origin of pride," says the prophet, "is to forsake the Lord, man's heart revolting against his Maker" (Ecclesiasticus 10:12 NEB). 


To expand on this idea with a bit of lexical precision, I will just point out that, although they certainly emphasize different aspects of sin, the Biblical writers recognized close connections between ḥṭʾ, pšʿ, and other terms in this semantic domain (most prominently ʿwn). Isaiah states this directly in terms of their common end (1:28):


  But rebels (pšʿ) and sinners (ḥṭʾ) shall be broken together,
  &nbsp;&nbsp; &nbsp; &nbsp; 
          and those who forsake the LORD shall be consumed.  


The Psalms include repeated rehearsals of synonymous parallelism using these two terms. This is often formulated in terms of Yahweh's grace, which, as with his punishment, applies "together" to every brand of evil entertained among men:


  I acknowledged my sin (ḥṭʾ) to you,
          &nbsp;&nbsp; &nbsp; &nbsp; and I did not cover my iniquity (ʿwn);
       I said, “I will confess my transgressions (pšʿ) to the LORD,”
          &nbsp; &nbsp; &nbsp; and you forgave the iniquity (ʿwn) of my sin (ḥṭʾ). (Psalm 32:5)




Notes  

 All English is ESV unless otherwise stated.   


1. There are, of course, many other terms. See, Trench, "Synonyms of the New Testament",  p. 239ff, for an extensive discussion. As pointed out by Silva (NIDTTE) and undoubtedly many others with modern linguistic sensitivities, the etymological distinctions drawn out there are not necessarily meaningful with respect to NT usage. Silva, regarding the dominance of ἁμαρτία + cognates:
  


  [U]sing the language of semantic hierarchy, ἁμαρτία may be viewed, with some qualifications, as the superordinate of the other nouns, which are its hyponyms... But more important, within a theological framework, ἁμαρτία becomes for the NT writers the key term to express the fallen human condition. 



2. Excerpted from NIDOTTE's index of semantic fields:  אָוֶן (mischief, iniquity, deception, H224); ‏חָטָא‎ (sin, commit a sin, purify, H2627...); ‏עָוָה (do wrong, pervert, H6390...); ‏עָוַל‎ (act wrongly, H6401...); ‏פָּשַׁע (rebel, violate, transgress, H7321...)
 


3. Like ḥṭʾ, pšʿ has both noun and verb forms that are both very common. Both also can be used as participles: "sinner", "rebel", etc. I have represented all of these with the root consonants here.
   


4. In fact, none of the common words for "sin" is used in the Gen 1-11 saga that is so formative for the Hebrew notion of sin, with the single exception of Gen 4:7 (as if we needed another source of obscurity in that verse).
  



References


"ἁμαρτάνω". New International Dictionary of New Testament Theology and Exegesis. Ed. Moises Silva. Zondervan, 2014.
  

 
Donald G. Bloesch. Essentials of Evangelical Theology. Prince Press, 1978; 1:93.
  

 
Eugene Carpenter, Michael Grisanti. "פשע", New International Dictionary of Old Testament Theology and Exegesis. Ed. Willem A. Wangemeren. Zondervan, 1992; 3:705.
    


Robin C. Cover. "Sin", Anchor Yale Bible Dictionary, Ed. David Noel Freedman. YUP, 1992. 



Alex Luc. "חטא", New International Dictionary of Old Testament Theology and Exegesis.  Ed. Willem A. Wangemeren. Zondervan, 1992; 2:90ff.