Either lute or small harp would work well - but I think you're better off with the harp because it's easier to get simple, nice-sounding melodies & chords out of if you're a beginner. Lutes take some technique-learning before you can get anything musical out of them, and I find them harder than guitar, though I speak as a harpist.

Any instrument is going to present a challenge if you want to play more demanding music, but for simple accompaniment the harp is great because it sounds good straightaway. Be warned, though, on the type of harp you're speaking of there will be no chromatics (sharps or flats). You have to tune into the key you want and then stay in it, and won't be able to alter notes while playing.

You can get cheap ones but they're usually crap. Where are you based, and how much were you thinking of spending? Get a decent instrument, even if it costs a bit more (which it will) because then you know it will work, it will last, and it will hold its value enough that you have something you can sell if it doesn't suit you. Best of luck - let us know what you decide -