The Idea in Brief
The use of this plural-form of the word appears to stem from its necessity to “fit” into the cantillation scheme of the second half of the verse. In other words, if the singular form of the word appeared, the short vowel (preceding the tone accent of the new word) would necessarily alter the cantillation pattern and therefore the dichotomy and meaning of the second half of the verse. This explanation of course would assume (along with the Babylonian Talmud in Nedarim Folio 37B) that the cantillation and vowel sounds were always inherent to the Hebrew Scriptures, although not codified in written form until later in the 9th and 10th Centuries by Masoretic scholars.
Discussion
Jewish Interpretations
First, there was no apparent plural meaning understood in this word by Jewish scholars. For example, the Targum Jonathan to the Prophets, which appeared as early as the 2nd Century, provides the simple singular translation of “one” without any nuance or innuendo of plural forms. For example, the highlighted phrase in yellow (below) contains the singular form of the word “one” in Aramaic.
Ezek 37:17 (Targum Jonathan to the Prophets)

Second, the medieval Jewish scholar Rabbi Shlomo Yitzchaki (“Rashi”), who lived in the 11th Century, made a similar observation, which was that the words should not be considered in any plural sense. The following translation of his commentary comes from http://chabad.org
Commentary of Rashi on Ezek 37:16
וְהָי֥וּ לַאֲחָדִ֖ים בְּיָדֶֽךָ: I shall join the two sticks, so that they will be one stick in your hand.
Jewish interpretations saw no plural meaning in this “plural” word. However, there appears a significant difference between the plural and singular forms of the word in this context, if and when viewed through the lens of accentuation and cantillation, and therefore meaning.
Significance of לַאֲחָדִ֖ים in the context
First, the received text according to the Biblia Hebraica Stuttgartensia appears as follows. The word in red is “long,” which, according to Wickes, means that this word must take its own disjunctive accent. This disjunctive accent must be the Tipcha.

When diagrammed in descending order of cantillation, the verse structure appears as follows.

Please note that the closing phrase, “in your hand,” modifies the phrase “they will become one.”
Also, please note that Athnach disjunctive accent, which divides the verse in half (the so-called dichotomoy) occurs at the fourth word from the end of this verse. According to Wickes, the appearance of the Athnach on the fourth word forces one dichotomy in the second half of the verse. So what if dichotomy were different? How would the meaning change?
Significance of לְאֶחָד in the context
According to the critical apparatus of the Biblia Hebraica Stuttgartensia, the suggested reading is the singular sense of the word. That is, the critical apparatus on this word in the verse reads as follows:
17 a prp לְאֶחָד cf 19 [ = “perhaps the word should be לְאֶחָד in light of Verse 19”]
If the “correct” word here were לְאֶחָד then the cantillation pattern would change, which then affects the dichotomy and meaning of the second half of the verse, and so the verse would appear as follows. Since the לְאֶחָד is not long, according to Wickes, this word would NOT be able to take any disjunctive accent, but would take the Merkha conjunctive accent.

When diagrammed in descending order of cantillation, the verse structure would appear as follows.

This arrangement would sound awkward because the verb here is intransitive, and like another example with an intransitive verb found in Exodus 19:16, the closing phrase, “one in your hand,” modifies the intransitive verb phrase “they will.” This separation of the intransitive verb from its immediate object seems more awkward than if the words just appeared together logically and syntactically by cantillation (as already noted above).
Conclusion
This explanation is not definitive, and is intended only to provide an alternative perspective on how the Hebrew Bible appeared and was heard (and understood) by Jewish scholars long before the system of cantillation marks and vowels were ever codified in writing in the 9th and 10th Century.
In this discussion, the “long” version of this word (לַאֲחָדִ֖ים) had enabled the dichotomy of the second half of the verse to occur later (before the last word of the verse). This arrangement provided the best meaning, since the dichotomy connected the intransitive verb with its immediate object.
However, if and when the normal “short” version of this word (לְאֶחָד) might have appeared, the dichotomy within the second half of the verse would have occurred immediately after the intransitive verb, and in this regard, would have separated the verb from its immediate object. This arrangement would not have been the ideal structure of this verse for the ears of listeners who heard these verses chanted through the system of cantillation, which was passed on through the centuries by oral Jewish tradition based on the original texts.
Reference:
Wickes, William (1881). A Treatise on the Accentuation of the Twenty-One So-Called Prose Books of the Old Testament. London: Forgotten Books, passim.