The Gayatri Mantra that is present in the Veda Samhitas has Varenyam in it and not Vareniyam.
The Gayatri mantra of Viswamitra Gatinah is not in the Gayatri metre but in a slightly variant metre called the Nichruth Gaytari which is made of 23 syllables and not of 24 syllables.
It is found in Rig Veda 3.62.10, Taittiriya Samhita 184.108.40.206 of Krishna Yajur Veda and in Vajasaneyi Samhita 3.35 of the Shukla Yajurveda. And in all these versions varenyam is only present.
tat savitur varenyam
All throughout North India varenyam is chanted. Probably they pronounce it as vareniyam in the South only.
The same Mantra is present in the Yajur Veda ( Vajasaneyi Samhita 36.3, Krishna Yajur Veda Taiitiriya Samhita 220.127.116.11 ) as well, but there the additional words are present viz
bhur bhuvah svah
thus changing the chandah (metre) to Ushnik from Nichruth Gayatri. Ushnik has 27 syllables.
I don't know how vareniyum came into being though. Probably something to do with Veda Shakhas ( ? ). For example, the persons belonging to Krishna Yajur Veda indeed replace the word Svah by Suvaha in the Mantra.
Abbreviations used in the answer :
Rig Veda 3.62.10 - Mantra 10 of Sukta 62 of Mandala 3 of Rig Veda Samhita.
Taittiriya Samhita 18.104.22.168 - Verse 2 in Anuvaka 6 in Prapathaka 5 of Kanda 1 of that Samhita.
Vajasaneyi Samhita 3.35 - Mantra 35 of Adhyaya 3 of that Samhita.
Another confusion which must be cleared is that Gayatri metre have to have 24 syllables. This is completely wrong. Gayatri is a name of a family of metres.
The number of syllables can vary from 26 to 19. The Gayatri metre has as many as 10 variants.. The Nichruth Gayatri is one such variant.
Apart from differences in the number of syllables, more variants occur due to the padas. Some variants of Gayatri have 5 padas.
Some of them have formations such as :
5|5|5|5|6, 5|5|5|4|6, 6|7|11, 7|7|7|, 7|6|6 etc and all of them belong to the Gayatri family of metre.
The Gayatri mantra that appears in RV 3.62.10 is having 23 syllables. Even in the classical reference, Katyayana's Anukramanika, Katyayana calls the 23-syllabled metre by the name Gayatri only.
Similarly, all the other major Vedic meters are having more than one variants. For example, Anushtup is having 11 variants. Brihati is having 11 variants and the Trishtup chandaha has as many as 14 variants.