The strange grammatical placement of this phrase is due to the Masoretic Text, which contains Hebrew cantillation marks that are both musical and logical. In this regard, the placement of the phrase בֶּן־אֲחִ֥י אַבְרָ֖ם (the son of Avram's brother) ends with the Tifha disjunctive accent, which modifies everything contained by the preceding disjunctive accent, which was the first phrase closed by the Tvir. To illustrate, please see the image below, and click to enlarge for better reading.
To recap, the first phrase of the verse וַיִּקְח֨וּ אֶת־ל֧וֹט וְאֶת־רְכֻשׁ֛וֹ (they took Lot and his property) is logically modified by the second phrase בֶּן־אֲחִ֥י אַבְרָ֖ם (the son of Avram's brother) based on the placement of the disjunctive accents. Also, the subsequent phrase וַיֵּלֵ֑כוּ (and they went) carries the second most powerful disjunctive accent in Biblical Hebrew, which is the Athnah. This particular disjunctive accent breaks the verse in half. The logical form of the first half of the verse therefore appears in the illustration below; please click to enlarge for better reading.
The second half of the verse is also parsed in similar divisions. The second half of the verse וְה֥וּא יֹשֵׁ֖ב בִּסְדֹֽם׃ (and he was dwelling in S'dom) ends on the most powerful disjunctive accent in Biblical Hebrew, which is the Silluq marked by the Masoretic sign of the verse end, which is the Sof Pasuq. Thus the second half of the verse closed by the Silluq and Sof Pasuq modifies everything captured by the Athnah, which (as already noted) is the second most powerful disjunctive accent. In other words, the second half of the verse modifies the first half of the verse (which is ALWAYS marked by the Athnah).
In summary, the division of words and phrases within the verses in the Scriptures was not only logical, but also accented in musical intonation. This hybrid of logical and musical arrangement of every verse not only helped to understand and clarify Scripture, but also enabled memorization, since both the logical and musical arrangement were denoted by the same cantillation marks.
Hebrew Cantillations (2014). Bellingham: Lexham Press.