Well I must reply to this post because my name is [name]Lyra[/name]! It is a very unique name and I love it! I admit when I was very young I felt a bit awkward at times having such an unusual name, but as I grew up I came to love it more and more. Now I cannot imagine having any other name. I love that people always remember my name, even after 20 some years (happens all the time when I bump into an old high school acquaintance, teacher or college professor). I am in my mid-30s and I have not met any other [name]Lyra[/name]'s my age. My parents chose it for similar reasons as yours. They were recently out of college when I was born, my mom a music major, my dad an amateur astronomer (and he still is). Since [name]Lyra[/name] is a constellation meaning the [name]Harp[/name] or [name]Lyre[/name], it fit the bill for the name they were seeking. My middle name is [name]Elizabeth[/name] and I have always liked how it goes with [name]Lyra[/name]. I think a classic/traditional name ties together with it well.
For upcoming generations I do not think [name]Lyra[/name] is that unusual at all. When I was a kid the name [name]Lila[/name] was not common at all, but now I hear it all the time. The two names seem quite alike to me, but I have to say I like [name]Lyra[/name] a little bit better (of course). 
I was never teased at all about my name when growing up. No one ever said anything about being a Liar, or Lycra. I do have to correct some on the pronunciation, as I get “[name]Lee[/name]-ra” sometimes. I just say, “[name]Lyra[/name] as in [name]Myra[/name], not as in the Italian currency.” I don’t think [name]Myra[/name] is that common of a name, but when I say [name]Lyra[/name] as In [name]Myra[/name] they seem to get it right away. The great part about the name, seriously, once someone knows it, they don’t forget it.
[name]Just[/name] before [name]Christmas[/name] I met a musician who has a 3 year old daughter named [name]Lyra[/name], [name]Lyra[/name] [name]Raelynn[/name] was her full name. It was so neat to hear of another little girl named [name]Lyra[/name]. I know in [name]England[/name] it is more popular than it is here. A lot of people here have heard of it because of the movie “The [name]Golden[/name] Compass”. It cracks me up hearing my name with the British accent in that movie.
For a middle name, I like your choices of [name]Rose[/name] and [name]Emmeline[/name] a lot. Since I’m used to [name]Lyra[/name] [name]Elizabeth[/name], I probably will tend to like [name]Lyra[/name] paired with a 3 syllable middle name. [name]Lyra[/name] [name]Emmeline[/name] is very nice. I always liked having a classic name as my middle name. My dad called me [name]Lizzy[/name] and still does as a nickname which I always liked too. As far as some other pairings that I think would go well a few that come to mind are…
[name]Lyra[/name] [name]Adeline[/name]
[name]Lyra[/name] [name]Cecily[/name]
[name]Lyra[/name] [name]Estelle[/name]
[name]Lyra[/name] [name]Genevieve[/name]
[name]Lyra[/name] [name]Gwyneth[/name]
[name]Lyra[/name] [name]Julianne[/name] (love this)
[name]Lyra[/name] [name]Juliette[/name]
[name]Lyra[/name] [name]Madeleine[/name]
[name]Lyra[/name] [name]Madeline[/name] or [name]Madelyn[/name]
[name]Lyra[/name] [name]Rosemarie[/name]
[name]Lyra[/name] [name]Violet[/name]
[name]Lyra[/name] [name]Vivienne[/name]
If you decide to go with [name]Lyra[/name] I don’t think you will regret it! I receive compliments on it all the time. One thing is for sure, your little girl will never have a problem with people remembering her name!
I really, really appreciate it! [name]Lyra[/name] [name]Elizabeth[/name] is just gorgeous, and I’ve thought of it, actually - the only problem in our case is that her last name will start with G, so I’d like to avoid having her initials spell “LEG,” “LAG,” “LOG,” etc, haha. It makes it a little trickier!