First of all, as mentioned in the comments, do not lead with the technology you want to use. If you want to move away from an existing system, regardless of what type of system it is, you have to justify its cost. The best way to do this, is to start with a cost/benefit analysis.
Gather information about how much time is spent using the current system, as well as a strong estimate of how much less time will be spent if the system is changed. You may not know the salaries of the people using it, but your boss will, and he/she can use the hours to come up with a cost associated to the time spent.
It is possible that time is not an issue, in which case try to attach a cost to the overbooking, and legibility issues that are side effects of the current system.
Once you have a dollar amount tied to the current system, figure out what a replacement system would cost for the same activities. If the savings is enough to justify investing in a new system, then you can start suggesting technologies to use.