The Coneheads wore little capes that just covered the shoulders.
What's the word or phrase for these little doodads?
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Since 1912 at least, they have been called 'capelets' according to Merriam-Webster online.
capelet : a small cape usually covering the shoulders
From the 14th century the word tippet seems to have been used. Merriam-Webster defines Tippet
2 : a shoulder cape of fur or cloth often with hanging ends
While I Marc Carlson's Glossary of some medieval clothing terms gives a lengthier description. (bolding mine)
[Origin uncertain; some suggest identity with OE. tæpped, tæppet, *teped (pl. tæppedu, tepedu) carpet, hanging, etc. = OHG. teppid, -ith, -it, tepid, -it: both ad. L. tapete (-a, -um) a carpet, tapestry hanging, bed-cover, table-cover. But there are great difficulties both of phonology and of sense. Others suggest a derivative of tip] 1. a. A long narrow slip of cloth or hanging part of dress, formerly worn, either attached to and forming part of the hood, head-dress, or sleeve, or loose, as a scarf or the like. Obs. exc. Hist.b. A garment, usually of fur or wool, covering the shoulders, or the neck and shoulders; a cape or short cloak, often with hanging ends. Now worn chiefly by women and girls, or by men as a part of certain official costumes. In many early quots. (omitted here), senses a and b are not distinguishable.
Since I believe this moment in the movie represents quasi-religious garb, I believe pellegrina is appropriate (although that is closely tied with the Catholic religion).
For the secular version, capelet or tippet is appropriate.