It is permitted to repeat words of davening for added intent in prayer. However, there is a virtue of having full intent the first time, and not repeating words. One can certainly say the word, and then reflect on its meaning, without saying it again.
The Mishnah (Berachos 33b) writes that one who says “modim, modim” is silenced by the congregation. The reason for this is suspected heresy.
The implication is that only one who repeats the word modim is silenced, and not others.
Furthermore, Rav Nissim Gaon and Tosafos quote from the Yerushalmi (Berachos 5:3) that the ruling applies only to the chazzan, and not for somebody who repeats the word in his private prayer, where it is considered a form of increased supplication. Some commentaries write that this distinction will not apply to the word modim itself. Yet, the Tur (Orach Chaim 61) notes that the distinction is not found in the Bavli, and therefore rejects it from halachah. The Bach, however, rules that the distinction of the Yerushalmi is valid.
However, even where there is no concern for heresy, it emerges from the Gemara and Rashi that a chazan must not repeat words, for this is disrespectful to the davening (see Rashi).
Several poskim thus write against the practice of chazanim repeating words of the davening just for the sake of the tune (see Aruch Hashulchan 338:8; Shemesh Marpeh Orach Chaim 4; Iggros Moshe, Orach Chaim 2:22; among others).
However, this problem applies when there is no good reason to repeat the words. If a person said words without proper intent, and he wishes to repeat them with proper intent, it is permitted to repeat words of davening (see Iggros Moshe, loc. cit.; Or Le-Zion 7:39; Hillel Omer 192; Pekudas Elazar, Orach Chaim 25; Eishel Avraham, end of Orach Chaim 51).
Yet, some authorities raise the issue of a hefsek, and mention that even for an individual it is disrespectful to repeat words. The Maharam Schick (Orach Chaim 31) thus writes against the practice of repeating words even for an individual (see also Yabia Omer Vol. 6, Orach Chaim 7:4).
There is therefore a virtue of having proper and full intent the first time, and not repeating words.