A couple of things about this post, as for remote data being incapable of being sent over a network, how exactly do the hundreds of thousands of headless servers out there actually work? Are they administered via psychic communication? They are administered remotely across a LAN or remotely via VPN/Intranet or over the internet and have been for thirty years, secondly who said anything about HDMI being the communication format?
That being said if you were to use DVI/HDMI and connect a remote monitor the cable length required before needing to amplify the signal is 25M or 50feet (reliably, I know of cable runs twice that length but the specification says that this is the maximum) without 'unacceptable' signal loss.
Option B: if this is only to be used in the short-term you could get an older laptop/netbook of somewhere like craigslist and use that with RDP
There are other options like KVM over IP, but they are more expensive that you wish to spend although a KVM switch with long cables, (DVI + USB) is another possibility. I have multiple PCs that I have connected via a DVI+USB KVM switch myself.
Most of the solutions I can think of would require cabling, unless you were to use a remote device (laptop/netbook/tablet) via wireless networking using RDP. There are tablets out there now with multiple OS support i.e. they come preinstalled with both Windows 7 and android, but once again there is a cost involved, these tablets are not the cheapest on the market, both Viewsonic and Lenovo of the well known brands have tablets with both operating systems preinstalled. That being said, the only real plus in this application of using a tablet would be the touchscreen, a netbook with it's own keyboard is an easier device to fly in the longterm and they are cheaper than similar spec tablets.
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