I would recommend H87/97 boards over H81 or B85, but if you must choose between those 2 go for an H81, Intel B motherboards have features that regular consumers will most likely never touch so it's not worth getting one if it costs more.
H87 has more robust features than H81 such as special Intel motherboard functions/mechanisms like Intel Rapid Storage and Smart Response Technology, more USB 3.0 slots, more SATA ports, more PCIE lanes, more Memory DIMM slots per channel, etc.
H87 should be your choice since it allows for more upgrades to be done to your computer, however, if you truly are on a tight budget an H81 will suffice, though it will have to be replaced if you would like to tinker and add to your hardware