‘Resolution’ is one of a group of nouns that come from the Latin word solūtus (meaning ‘made loose’) and are all related to verbs ending in ‘-solve’ (from the Latin solvō, meaning ‘I release’).
Other members of this group are ‘dissolution/dissolve’ and ‘absolution/absolve’. They all suggest some sort of release or breaking down of a substance into a simpler state. In the case of ‘resolution’ the ‘re-’ part indicates intensity rather than repetition. So a resolution is a ‘complete breaking down’ rather than a ‘breaking down again’. This makes sense when it relates to the resolution of a problem or a dispute, but what has it got to do with the promises we make to ourselves at the start of the year (and have usually broken by the end of January)?
The answer is that the word has acquired various extended meanings over the course of the centuries. From its basic meaning of ‘complete breaking down’ we get another sense of ‘revealing the details of’, and this in turn leads to the meaning of ‘a formal decision to do something’, which is the meaning that we think of at New Year.