I'll assume that by "have the last state of the corresponding output of the TXB0108 preserved" that you mean "... preserved on the non U-controller side of the TXB0108", and that only one line is involved ad that it is always the same line and that it is mono directional and outgoing from the u-controller. And that .. :-).
An adhoc one line unidirectioal "bus-holder" can be made with a non inverting buffer with a resistor from output to input OR
2 x digital inverters in series (to form a non inverting buffer) with a resistor from output to input.
Using inverters is more common as they are normally more available as part of an overall design - but with hex buffers you'd get twice as many per package. Using Schmitt triggered devices is nicer but not essential. This is essentially a level driven latch.
The first diagram below is simply a non latching inverter.
The second diagram shows a non inverting buffer with the 3M resistor providing positive feedback for latching. The 1M adds Schmitt trigger action and can be omitted when used as described. The actual value of the '3M' resistor can be set to suit the situation and logic family. Lower gives a stiffer latch to "pull against". Higher reduces power and makes toggling easier.
The third diagram shows the two inverter version. An advantage is that invered and non inverted latched outputs are available. Again, the 1M adds Schmitt action and is not needed here. The inverters can be Schmitt or non Schmitt types. (CD40106, xx74yyC14, ...)
A bidirectional version can be easily implemented subject to certain limitations. he 1M resistor is omitted and the circuit is driven at its "input". While the drive level exceeds the feedback resistor + inverter drive the input line drive dominates. When the input line goes open circuit / high impebance the lacth self hold via the "3M' resistor and the input line is "latched" by this as well. Making the resistor lower in value provides more drive. For example, if the drive circuit can easily overcome inverter drive via a 10k resistor then using a 10k provides that level of drive impedance when the input goes to high impedance.
A circuit can be design tailored to an actual need if the need is fully known and ICs to do yur task are probably available as noted by @markrages.
Above circuits from beam-wiki on Schmitt triggers
Have another go at the numeric hint. The question told you they were binary values.
We normally write binary values left to right, same way as we do for decimal values. So the left hand column is most significant, so weight 4, and the right most column is least significant, weight 1.
You understand XOR gates, right? One way to look at them is 'they do the exclusive OR thang'. Another way to look at them is that they pass through a logic signal unchanged when the other input is 0, and invert it when the other input is 1. The logic operation described is identical, but sometimes it's easier to get what a circuit is doing using one or the other way.
If you look at the logic operation, a pattern emerges.
The '1' output is always inverted.
The '2' output is inverted when the '1' input is low.
The '4' output is inverted when both the '1' and '2' inputs are low.
That sequence is screaming out a pattern to me, like it's meant to do something sensible.
Anyhow, I'm not going to do your homework for you. You started in the right way, writing down all 8 sets of ones and zeroes. However if the answer didn't jump out at you, perhaps you picked a non-standard weighting. Try again with the weighting I suggest and have another look.
Best Answer
Try to use JK-flip flop with two AND gates. J and K input connect to Vcc and AND gates place on the inverted and non-inverted output. The outputs change everytime the input is one and when the inputs is 0 the AND gates turn off the outputs. Probbably the simplest solution.