9V instead of 3.7V 8.14WH
In short, no. The solar charger (however it is constructed) can be assumed to output a fairly constant voltage (assuming same sunlight) of around 3.7 to even 4.2V. This is sufficient to charge the battery it came with - which is a standard value (3.7V) for lithium rechargeable batteries. If you connected a 9V, not even considering how you might harm the solar charger, it wouldn't even be able to charge it past 4.2V, and that's putting it generously. The max you might get is 3.7 to 4 volts on the 9V battery.

Remember how sources work in parallel - this is how things charge. A 5v battery in parallel with a 3 V battery will work to charge the 3v up until they are both at an equivalent potential. If you did connect the 9v battery, it wouldn't work very well, and could harm the solar charger if there isn't a diode/other protection means in place.

I would recommend you replace the battery with the same type. Going off of an assumption from your post I will take a guess and say it uses some sort of lithium cell, like a 18650. These are plentiful and would work properly. You should buy one of those (at a similar capacity, if not same) and replace it - it would work like new.

You just have to search '18650', or rather any other lithium cell of 3.7V and a similar capacity, swap it in, and it should work. You might want to give it an initial charge first though (there could well be a circuit in the solar charger that stops charging devices if the battery is too low).