I'm struggling to turn up any answers on why the word スタンド is used to mean "desk lamp". My guess would be that it's a case of metonymy, but I don't know.
1 Answer
It's short for 電気【でんき】スタンド, which we can see in sense ② in Daijirin:
②「電気スタンド」の略【りゃく】。
Here, 略 means "abbreviation". And of course, 電気{でんき} means both "electricity" and "[electric] light". Why does 電気 have this meaning? Well, most dictionaries don't say, and I suppose this could be simple metonymy, but when we look up 電気 in 日本国語大辞典, we find:
③(「⇨でんきとう(電気灯)の略)電灯{でんとう}。
So we could consider スタンド as indirectly short for 電気{でんき} "electric" 灯{とう} "light" スタンド "stand".