I just want to film up to three 5-min runs per ride, then upload them to laptop and wipe the memory card each time. What’s the smallest memory size card I can get away with?

Why worry about the smallest card? An 8GB should be fine but for a few extra ££’s you could get a 32GB which would mean you can record more if you want to. Also depends on the quality you’re recording in..
edit: I got a 32GB class 10 from Amazon for under £13

As above, they’re so cheap just get a 32.

32gb does approx 2 hours at 1080p on my Gopro3.
You can pick up a 64 GB for £25 now, so I would not worry about buying the smallest possible.
Go for a fast card (40+mbps) and maybe get one that has a USB3 reader with it unless you already have that.

It’s not just capacity that you need to consider but speed, some cards are faster than others and video is pretty intensive so you are better getting faster cards. Card speed is defined by class numbers, you will want class 4 or 8 if you are shooting in HD, class 2 is good enough for standard definition.