I don't think the hardware should make a tangible difference in terms of security if you're comparing Raspberry Pi with a standard 32/64bit platform (i386, amd64 etc). Unless you consider physical security (which you don't), or perhaps from availability perspective, which is also an important part of security, but I'm guessing this is not what you're after.
From what I see about the Raspberry Pi, it's simply an ARM platform, typically used in many devices like smart-phones, routers, voip gateways etc. ARM is a very widely deployed platform, so in that respect it's still a good choice. It's definitely not obscure.
Whilst there might be some marginal differences in terms of what security features the platform can offer. For example whether or not it uses the NX bit, I would say that your primary concern is not the hardware but rather the software that you're running, and the way you expose your server to the outside world. Software itself might be vulnerable on either platforms and this is more likely to be your "weakest link". This includes the software you choose (OS, web stack, services, listeners etc) or your own application if you develop it.