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Choosing a yarn for a blanket is a very personal choice, dictated by things such as your budget, how many colours you're after, desired warmth, etc. In the end it's up to you, but below I've given an outline of how different yarns might affect the final look and feel of your blanket. Then I've described the yarn I chose myself, and finally you'll find some suggestions for alternatives.
Cotton
Cottons give you great stitch definition (which means neat squares) and are machine washable, so they're a practical choice. They come in a wide range of prices, so you can usually find one to suit your budget. Cottons are a good choice for baby blankets and mid-season to summer throws. However, they don't give much warmth so wouldn't be suitable for a winter blanket. They can also feel quite heavy in spite of not being warm, so I wouldn't go for anything beyond a DK yarn.
Wool + Cotton
A 50-50 blend of cotton and wool will give you the stitch definition of the former, but with the added soft touch and a little more warmth thanks to the wool. Most brands use merino, and they can usually still be machine washed. Prices tend to be mid-range.
Wool
Wool means softness and warmth! If you're after a blanket for winter, a smooth wool such as merino will give you both stitch definition and warmth. A "hairy" yarn such as alpaca will feel wonderfully soft, but the stitches won't look as crisp. As wools are usually lighter than cotton, however, you could use anything from 3-ply to chunky. Prices vary from mid-range to very expensive.
Wool + Silk or Cashmere
Beyond doubt, the luxury choice. If you want an heirloom blanket that has everything - stitch definition, warmth and lightness - a blend of wool and silk or cashmere is the way to go. Obviously, it's not the budget-friendly choice but could be appropriate for a very special baby blanket.
Acrylic
The overwhelming advantage of acrylics is that they are cheap, allowing you to make a large blanket at low cost. However, they generally feel quite stiff, don't give much warmth and don't always wear well. There is also their environmental impact - you can buy organic cottons and wools, but there is no such thing as organic plastic...
For all of the squares and border in the Crochet meets Patchwork Blanket, I have used a yarn called Natura Just Cotton by DMC. It's widely avalaible from craft stores as well as online yarn shops.
Details
What I like about it
What I don't like about it
Which cotton yarn could I go for instead?
NB: This list just contains some suggestions and is by no means exhaustive!
Budget
DROPS Muskat or Paris - widely available, good range of colours, reasonable soft handle.
Mid-range
Vinnis Colours Nikkim - amazing range of colours, very soft for a cotton, hand-dyed in South Africa.
Nurturing Fibres ECO-Cotton - organic, soft cotton with a high twist. Doesn't split!
Debbie Bliss Eco Baby - colour range is limited but very pretty. Fairtrade.
Luxury
MoYa Whisper or Harmony - organic, very soft cotton that comes in a beautiful colour palette.