Is she dissatisfied?
Is she upset by this situation? Does she desire greater involvement, or would enjoy more if there were something different about the game? Or does she like her low level of involvement?
There is no way we can answer these questions. They are questions for her. Talk to her, ask her straight out. She may be just fine with things the way they are; her level of involvement may be all she wants. She may prefer to be there for the camaraderie and social interaction, to be a part of the group, more than for the game itself. Or, there may be very different things she wants from the game, which she doesn’t know how to bring up or is too shy/polite to do so. Those are important things to know.
Here, I strongly recommend the Same Page Tool. It’s a great way for groups to get, ahem, on the same page. It will help frame the question of what she wants and what the game is actually providing and how those things differ.
If she’s comfortable: does it negatively affect anyone else?
If she doesn’t want more involvement, is that acceptable to the group? Or does the character who is apparently silent and uninvolved most of the time occasionally piping up cause breaks in immersion?
Again, questions for you and your group. I, myself, for whatever reason just did not get very involved in a game recently. As a group, we decided it was OK for the character to “check out” most of the time, with me only adding in quirky one-liners and tossing out the occasional spell in combat. It fit the character, and the group was OK with it.
In a different game with a different group and as a different character, I may not have gotten away with that. I most likely would have dropped out graciously, since I just wasn’t getting into the game. She may as well, or she may agree to some compromise, to try harder to get involved.
If she does want more, can you reach a compromise that will give her more opportunity to engage?
If there are things she wants from the game that she’s not getting, can you offer more of them without disrupting the game? Or are they incompatible with the game that you and the rest of the group want to play?
Once again, questions for you and your group. Be upfront and honest here; no one wants to waste time on a game that is frustratingly not providing the experience you want, and it won’t improve your game. But if possible, assuming she is a friend/you enjoy her presence in the game, compromise is a good thing. It’s entirely possible that what she wants is easy enough to provide, or a good idea in any case, or something the rest of your wants as well unbeknownst to you.
Yes, it can
You can make any non-magical trinket that can fit in your hand, and it disappears after the end of your next turn. One of the trinkets you can make, as listed under PHB 160, is "a whistle made from gold-colored wood" as well as "a tiny silver bell without a clapper"; these are both things that you could make such that they can fit in your hand.
Furthermore, they fit the criteria of "musical instruments," because you can make music with these. While the PHB does list some of the things it considers as instruments, it is not meant to be an exhaustive list:
Tools, PHB 154
Musical Instrument. Several of the most common types of musical instruments are shown on the table as examples.
This means prestidigitation can create a musical instrument that can fit in your hand, and you can use any musical instrument as a spellcasting focus.
Of course, it is inefficient
You must use your action to cast prestidigitation, and then you cannot use your action again to cast another spell. On your next turn, you could use the instrument to cast a spell, but then it disappears at the end of that turn, and you have to recreate it on your third turn. This allows you one spell every other round that isn't prestidigitation (unless it's a bonus action or reaction spell).
Best Answer
The DM is charged with making rulings on a huge variety of things that go on in the course of playing the game. You can make your case for why you think it should be a given way, and then await a ruling.
Once the ruling has been made at the table, the DM is right1.
During play, accept that and then press on as the other players wish to play for fun and are probably not there to watch an argument.
Once play is done for that session, you have reached a potential decision point.
Is this particular decision a deal breaker for you? Do you want to revisit it outside the time constraints of a gaming session, in a non-confrontational manner? If not, if mostly you are having fun, then roll with it.
In this case, you picked that cantrip with the expectation that you could do certain things, and have just found out that you can't. In a non-game time environment (or in a friendly email) present the PoV that your expectations were not met (by accident) and ask for another cantrip. Or, ask that she reconsider the ruling once you've explained your position. Your request must be unemotional and non-confrontational.
DMs do this for fun, not for pay.
Once you've re-stated your case, accept any follow-on ruling with good grace, pro or con. She does a lot of work to run this game for your group.
Peace between you two will benefit the whole table.
If this ruling is a deal breaker ... if this is a decision in a pattern of rulings that you find are consistently dashing your expectations against the rocks ... then you need to have a different (unemotional) dialogue with the DM:
Are you two playing the same game with the same expectations?
If you can't reconcile that, this table may not be a good fit for you.
Even if you don't always see eye-to-eye, there are some things that you can do. You can contribute some good faith effort to future decision points.
Without going through the eye-watering detail of the Same Page Tool, it's worth your while to look at spells that you are interested in adding to your spell book before you get to them. Likewise the spells you already have.
Do some homework, and a little forecasting. Try out some practice scenarios yourself. What would I do in X case? In Y case?
In an email or a conversation outside of a game session, present some of your ideas on how a spell might work beyond its obvious uses. A lot of DMs appreciate ideas raised and resolved before or after the gaming session where a non time-critical decision is achievable. (I sure did when I was running games during my early DMing years, and I really appreciate it now that I am running them again. My group has a lot of "how does this work?" and "I thought it worked like this" episodes).
You may be able to get her to "see it your way" a few more times when she's not under time pressure. Or not. However this dialogue plays out will inform your decision on whether this is a good table for you, or not.
Insofar as the interpersonal skills: the more non-confrontational you make your approach to her, the more likely you are to get her to see it your way on some issues, but you won't always get your way.
Why?
At the end of the day, the DM's rulings are the rule. She's got more than your fun to consider, she has the whole table's fun in her hands as DM.
Best wishes for continued fun at this table, or any other.
1 (PHB, p 6)