Cheshirization, or cheshirisation, is a type of sound change where a trace remains of a sound that has otherwise disappeared from a word. The term was coined by James Matisoff and is a neologism, i.e. it is not an established scientific term. It is used here to refer to a process that is real but so far has no generally accepted name. The term rephonologization has sometimes been used to refer to this process; see below.
Essentially, a distinction between two sets of words that was formerly expressed through one phonological feature (e.g. a particular sound) is preserved (or partly preserved) through being re-expressed using a different phonological feature. This typically occurs through two sound changes: one that introduces a modification of some sort, conditioned on the presence or absence of a particular feature, followed by another change that deletes or changes the conditioning feature.
A common example is Germanic umlaut. In many Germanic languages around 500–700 AD, a sound change fronted a back vowel when an /i/ or /j/ followed in the next syllable. Typically, the /i/ or /j/ was then lost, leading to a situation where a trace of the original /i/ or /j/ remains in the fronted quality of the preceding vowel. Alternatively, a distinction formerly expressed through the presence or absence of an /i/ or /j/ suffix was then re-expressed as a distinction between a front or back vowel.
As a specific instance of this, in prehistoric Old English, a certain class of nouns was marked by an /i/ suffix in the (nominative) plural, but had no suffix in the (nominative) singular. A word like /muːs/ "mouse", for example, had a plural /muːsi/ "mice". After umlaut, the plural became pronounced [myːsi], where the long back vowel /uː/ was fronted, producing a new subphonemic front-rounded vowel [yː], which serves as a secondary indicator of plurality. Subsequent loss of final /i/, however, made /yː/ a phoneme and the primary indicator of plurality, leading to a distinction between /muːs/ "mouse" and /myːs/ "mice". In this case, the lost sound /i/ left a trace in the presence of /yː/; or equivalently, the distinction between singular and plural, formerly expressed through a suffix /i/, has been re-expressed using a different feature, namely the front-back distinction of the main vowel. This distinction survives in the modern forms "mouse" /maʊs/ and "mice" /maɪs/, although the specifics have been modified by the Great Vowel Shift.