In Maoz Tzur we say:
רב בניו וקנייניו,
על העץ תלית
His many sons and his possessions,
You hung on a tree
Were Haman's possessions really hung on a tree?
It's amazing what you can find on Hebrewbooks!...
In Sifsei Chachamim, by R' Avraham Abba Hertzel (Bratislava, 1899), he says that this is based on the Gemara's statement (Megillah 15b, top) that "that wicked man had all of his treasures engraved on his chest" (evidently meaning that he wore a medallion, or something similar, that had all of his possessions depicted or listed on it). Presumably he would have worn this to the two feasts to which Esther invited him, and since he was taken directly from the second one to be hanged, then that medallion - "all his treasures" in microcosm - was hanging there on the gallows with him.
At the 2014 International Bible Contest for Adults (חידון התנ"ך הבינלאומי למבוגרים תשע"ה) televised finals1, celebrated grammarian2 Dr. Avshalom Kor (אבשלום קור) posed this question among a series of short vignettes about "Ma'oz Tzur" that he presented while the next contestant was getting into place. He answered that the 'ו' preceding "his possessions" is an example of a "ו' שמסבירה" - a "vav that explains."3 So, this 'ו' is more of a dash than an "and."
Dr. Kor provided, as a Scriptural example of this construct, Shmuel I 28:3, which says, of Shmuel:
... וַיִּקְבְּרֻהוּ בָרָמָה וּבְעִירוֹ ...
... and buried him in Ramah, וּ in his own city ...
The 'וּ' here doesn't mean that they buried him twice, in Ramah and in his own city, but that they buried him in Ramah, which was his own city.4
Similarly, here, the song says that they hung up Haman's sons, who were his possessions, not as well as his possessions. This explains the translation choices of siddurim documented in Yosef's answer.
1. In which one Alexander Heppenheimer, 42, of Crown Heights, took second place.
2. Yes, according to Wikipedia, Israel has such a thing: "עקב קנאותו לשפה העברית וסלידתו משגיאות כתיב וטעויות דקדוקיות בדיבור הפך שמו לשם נרדף ליודע השפה העברית והלצות רבות על הדקדוק בעברית נקשרו בשמו."
3. Or, as seen in footnote 1(b) to Gesenius, §154a, wāw explicativum. (Thanks to msh210 for the reference.)
The translations I have seen translate it differently, and effectively elide the vav; either קניניו refers to the rest of Haman's household, or to the fact that his sons were his dearest possessions.
His many sons and his household You hanged on the gallows.
His numerous progeny -- his possessions -- on the gallows You hanged.
Der Söhne Schar, sein teurer Schatz, an seinem Galgen aufgeknüpft.
which means: "His multitude of sons, his dear treasure, were hung on his own gallows."
Hans dyraste egendom, hans många söner, lät du hänga i galgen.
which means: "His most precious property, his many sons, You let hang on the gallows."
In Riv'vos Efrayim (volume 8 number 267), Rabbi Efrayim Greenblatt suggests that it may refer to Haman's slaves. (He also refers the reader to Or L'avraham on Rus, by Rabbi Avraham Gurewitz (spelling?), page 98; but I don't have a copy.)