What is the most accurate and cheap way to figure out the distance between two points electronically?
I have been envolved with the design of commercial systems that were either for the purpose of outright reporting a tag's position, or it needed to know that as part of a larger scheme.  Among these systems there was use of IR, RF, and ultrasound.  I can tell you that none of them solves the problem "nicely".

RF can be used to find location to a reasonable resolution based on triangulating with signal strength.  However, various things mess up the received signal strength.  If you go with RF, use a low frequency compared to things like WiFi and many other RF systems.  We used 434 MHz, which is one of the ISM frequencies.  On that band it is permissible to send a short message of a limited power once every 10 seconds unless the user deliberately initiates some action.  Higher frequencies get diffracted and obstructed more in a indoor enviroment.  Then there are issues of varying received signal strength due to polarization.  There are ways to deal with that, but that would be a whole discourse on its own.

You might be tempted to try to find location by measuring time of arrival - at least until you do the math.  At only a nanosecond per foot, the multiple receivers have to be very accurately synchronized.  Even more difficult, they have to be able to determine some common event in the transmission to the resolution of a single carrier cycle, but of course the bandwidth limitations will only allow small changes between adjacent carrier cycles.  Basically, if you have to ask here, RF time of arrival is way over your head and probably your budget.

IR does basically what you want, except for the fact that it is easily blocked by your body and clothes.  It's not going to work in a pocket.  It can work clipped to the shirt, but not if it gets covered by a lapel or a sweater or something.  IR will generally bounce around a room nicely, but not much power makes it thru the relatively small doorway.  If you can deal with the occlusion problems, it is a pretty good way to localize a tag within a room, since walls are opaque to IR.

Ultrasound also won't go thru walls, but bounces around a room less well.  It can go thru a thin layer of clothing sometimes and sometimes not.  There is also a lot more ambient ultrasound noise than RF or IR.  Even if the tag is worn on the chest and is not covered and the transmitter is facing out (it's real easy to have a tag flip transmitter side to the chest), there is still a reasonable chance of a message not getting thru.  You can do useful time of arrival triangulation with ultrasound since it is so slow (takes about 3 ms/m), but that also means the data rate is very slow.  If these tags can emit a single ping, then you don't care about data rate.  If you need to identify multiple tags, then you have to send some sort of information with each message.  We ended up encoding information in the timing between short bursts.  But, keep in mind that there is a inherent lower limit on time between bursts.  You can't send a new one until the last one is done echoing around the environment else receivers will get confused.  In practise, you need to wait for about 10 meters of propagation for things to die down between bursts.  The customer may have filed for a patent on this, I'm not sure.

In any case, you have to consider the channel from tag to receiver as being error prone.  If you attempt to send any data at all, it had better include a checksum.  We actually ran into trouble with a early RF system that used only 8 bits of checksum per packet (I didn't design that packet format).  There was enough bad stuff getting interpreted as good data to cause field problems.  The newer packet format (which I did design) had a 20 bit CRC checksum, and that basically took care of the problem.