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When thinking of short slogans or sayings there is great value in having something that is fun to say and has good shape, but not necessarily directly rhyming. If the rhyme is too literal, it tends to have a sing-songy tone.
The phrase “Keep calm and carry on” felt as though it had the property I’m looking for.
Is there a term to describe this kind of “almost-rhyme”?
There is no rhyme in the sentence, but its rhythm is iambic, that is, one unstressed syllable followed by a stressed one. It is typical of English speech and frequently found in English verse. The sentence is also alliterative, in that three of the five words begin with the sound /k/. Alliteration was a feature of Old English verse, and it still appeals to the ears of English speakers. It may be these features combined that you, and perhaps others, find pleasing. It has to be said, though, that the sentence, which has its origins in Britain in the Second World War, has been done to death.
Near rhymes (which also go by several other terms, such as inexact rhymes or slant rhymes) are words that do not completely rhyme, but parts of them do, or the words sound similar. I think "Keep calm and carry on" is more of an example of consonance:
Consonance is a poetic device characterized by the repetition of the same consonant two or more times in short succession, as in "pitter patter" or in "all mammals named Sam are clammy".
Consonance is not just for poems, however. It also gives prose a pleasing, memorable cadence.
Perhaps you are looking for the word "alliteration"?
