יח וַיְהִי בְּצֵאת נַפְשָׁהּ, כִּי מֵתָה, וַתִּקְרָא שְׁמוֹ, בֶּן-אוֹנִי;וְאָבִיו, קָרָא-לוֹ בִנְיָמִין.
18 And it came to pass, as her soul was in departing--for she died--that
she called his name Ben-oni; but his father called him Benjamin.
The traditional understanding is that "Ben-oni" means "son of my suffering," and Yaakov renamed him to be "Ben Yamin," "son of my right-hand," meaning "son of my strength."
However, perhaps we can treat אוֹנִי as "my strength" as well, with Rachel calling him "son of my strength" in Aramaic and Yaakov in Hebrew.
We see Yaakov uses אוֹנִי to mean strength. When he tells his sons what will happen to them in the "end of days," in Bereishit 49:3, he says
רְאוּבֵן בְּכֹרִי אַתָּה, כֹּחִי וְרֵאשִׁית אוֹנִי--יֶתֶר שְׂאֵת, וְיֶתֶר עָז.
Reuben, thou art my first-born, my might, and the first-fruits of my
strength; the excellency of dignity, and the excellency of power.
Additionally, this might be a dialectal difference - we must recall that Rachel was from Charan, and the daughter of an Aramean, so she might give an Aramaicized name in his name, where as Yaakov would speak Hebrew, and would give the Hebrew. (If so, it is an Aramaic cognate Yaakov uses in Bereishit 49)
Something akin to this is in fact suggested by the midrash rabba on Vayishlach. They say that בֶּן-אוֹנִי is בר צערי, the son of my affliction, in Aramaic, and Yaakov gave him his name in Hebrew. Thus a linguistic difference based on place of origin, but also a different meaning. (One perush brought down in the likut on the side suggested, citing eventually Rashi, that they both mean strength, but she meant the strength of the difficulty of her childbirth, and Yaakov spun it in a more positive direction, strength in general. I would note that, with a bit of forced reading (perhaps made easier if there were some variant text) the midrash might be parsable to take the בר צערי as one explanation, and the Aramaic/Hebrew variation as a second perush, in which case it would be exactly what I initially suggested above.)
Update: My wife Racheli pointed out the parallel to the naming that occured in the previous parsha, Vayeitzei. Lavan pursues Yaakov, and in the end they make peace. We read, in Bereishit 31:46-47:
Lavan, Rachel's father, is an Aramean, so he gives the heap of stones an Aramaic name: Jegar-sahadutha = "heap of witness" in Aramaic. Yaakov speaks Hebrew so he calls it Gal Ed = "heap of witness."
Thus, the are calling it the same thing, but in different languages. The structure of this pasuk is remarkably similar to that of the pasuk in which Binyamin is named.