How much free space on my phone is needed to install Windows Phone 8.1 Developer Preview?
Windows Phone 8.1 is a massive update to the OS and hence will require quite some space to download, extract and install the update. Keep as much free space as you can, but minimum should be around 1 GB.
Storage Sense reports that on a Nokia 625, The system takes up 4.21Gb after installing the Nokia Cyan update (which includes a couple of Nokia specific bits on top of the update to 8.1) - this phone needed in excess of 2Gb free to download, unpack and install the update.
If you do need to free disk space, you can uninstall any apps without incurring any additional charge to reinstall later; for example, you can safely uninstall Here Maps or Here Drive, and reinstall after the update - be careful with apps where you have in-app purchases for "consumable" items, as the unused portion might not carry through (I've not tested this scenario).