The usual term is just rhyme! (In sense 1c.)
rhyme, n. 1.b. Correspondence of sound between the endings of two or more words or metrical lines such that the syllables involved carry identical vowel sounds and have (if present) identical final consonants.
c. An instance of such correspondence of sounds; a word so corresponding in sound with another.
Oxford English Dictionary.
Where there may be confusion with the more abstract sense 1b, some authors use rhyme word:
1896 The Athenaeum no. 3569 p. 372 The poet must know the instant that a rhyme-word presents itself to his mind what rhyme-word will do swift and concise work, and what rhyme-word will not.
1902 Hartley Coleridge Works of Lord Byron vol. VI p. 278 The repetition of the same rhyme-word was noted in Blackwood’s Edinburgh Magazine
1925 Amy Lowell John Keats vol. II p. 204 It may be objected that “moors” is also merely a rhyme word, but this I doubt, principally because it does not properly rhyme at all.
However, the part of the line that rhymes is sometimes shorter than a word:
At mortal batailles hadde he been fiftene,
And foughten for oure feith at Tramyssene
Geoffrey Chaucer (c. 1400). General Prologue to the Canterbury Tales, lines 61–62. Wikisource.
and sometimes longer:
There stands the Master. Study, my friends,
What a man’s work comes to! So he plans it,
Performs it, perfects it, makes amends
For the toiling and moiling, and then, sic transit!
Robert Browning (1855 rev. 1868). ‘Old Pictures in Florence’. In The Poetical Works of Robert Browning, volume III, p. 134. London: Smith, Elder.
Perhaps for this reason, the term rhyming element is sometimes used for the sound or group of sounds that correspond: “ene” in Chaucer and “ansit” in Browning.
Normally the last stressed vowel in the line and all sounds following it make up the rhyming element.
Chris Baldick (1990). The Concise Oxford Dictionary of Literary Terms, p. 189. Oxford University Press.