Can I use an amplified splitter outdoors on a aerial antenna?

I'm not clear with what you are trying to do or why.

You say that her antenna already has an amplifier on it that is helping and I am assuming this amplifier is intended for outside operation, unlike the one you have come up with. So what are you trying to accomplish here by moving to one that is not even designed for outside operation? Why not keep the old one?

Also, the "newness" of an antenna has nearly nothing to do with how good of a signal it puts out (assuming the coax and connections to it are in good shape).

There are two things that have to be considered for any TV antenna. It's gain and it's frequency range.

You can look at an antenna and the number of elements it has to get a rough idea of which antenna has a higher gain. It is as simple as the antenna with more elements is going to have the higher gain.

I'm guessing that she has several TVs in her house and that at least one of them isn't getting a good signal and this is what you are trying to deal with. My approach to solving this would be to go back to square one here and look at the entire system and them change the absolute minimum that you need to change in order to solve the problem. Your approach is more of a shotgun approach.

First, lets take a look at her location versus the stations she gets or wants to get. Start here:

http://www.antennaweb.org/aw/welcome.aspx

This will give you an idea of the type of antenna she needs, lets say "medium directional".

Now you can go to http://www.channelmaster.com/ and or http://www.antennacraft.net/ and see what their antenna recommendations look like and compare those with the antennas she has.

You may find that one of the antennas she has looks very similar to what they suggest. If so, I'd keep that one. Then you will have to look at the direction to each of the stations she wants. If she is close enough in or if the stations are all about the same direction from her, she may not even need an antenna rotor, but otherwise, she will.

Now take a look at the coax she has on her antenna. If it is older, it might be the smaller diameter RG59 instead of RG6, which has a lower loss. If the coax is in good shape and the runs are short, it may not make much difference, but if either of those are not true, new coax will make a big difference.

Then consider the number of TVs she is trying to run. If she is just running 2, splitting the signal in half will knock 3 db off the signal strength. That usually isn't much of a problem but if there are more splits that that, I can see why the amplifier is being used. However, since you say the current ampler is working, I really do not understand your push to replace it with something that is newer, but something that is designed for inside operation only. Even if you put it in a box, in most areas of the country, you can have some pretty moist weather with rapid temperature changes, which is going to mean you are going to have moisture condensation inside whatever box you supply and whenever you have moisture, you have a potential for problems that I would not want to put my grandmother through.