A guitar-like fretted, stringed instrument that you play on your lap. And not a lap steel, but something that takes full advantage of the way it is held. The orientation of the instrument would allow for the addition of many more strings, spaced farther apart, and a larger/differently shaped soundbox without concern of being too heavy or unbalanced in weight. Moreover, one could look at the fretboard while playing and use the full dexterity of the left hand, allowing piano levels of playability. As someone who has dabbled in piano and guitar, I miss the expressivity of manipulating the strings myself when playing the piano, and I miss the elegance and open-endedness of the keyboard when playing guitar. So something like this would be my dream instrument.
There is something like what you described and it's called harpejji. It's not placed on your lap, but on a stand in front of you, but it is pretty much what you described:
The instrument aims to bridge the gap in sound and technique between the guitar, bass guitar, and piano. The playing surface has an isomorphic keyboard layout arranged in ascending whole tones across strings, and ascending semi-tones as the strings travel away from the player with a five octave range from A0 to A5. Wiki
It basically looks like a guitar fretboard with more strings, that you play like a piano.
You might have seen it from Jacob Collier:
Aside from kids' toys, there are the zither and the autoharp; the latter being a zither with chording capabilities.
Moving over a bit, there's the hammered dulcimer.
Moving over a lot, there are many Chinese traditional stringed instruments which are played horizontally. You can find variants with 4 to a couple dozen strings.
While you said "not a lap steel" in your question, I did want to make sure that you were aware of the console steel guitar and the pedal steel guitar. These two instruments are significantly evolved from their "lap steel" roots, and fulfill two of your main criteria: they have many more strings, and they allow you to look at the fretboard while playing & use additional left-hand dexterity. However, as electric instruments they do not have a soundbox per se, and they are too large to actually be played on the lap; rather, one sits or stands at them like a piano.
(Both images taken from Wikimedia Commons)
Except for the small number of strings an Appalachian dulcimer seems close to matching what you want. But, really it seems like the harpejji in @Shevilaskovic's answer must have been invented for exactly the hybrid you're looking for!
Also, you may already know about the mechanism for the clavichord, but in case you don't it has hammers that strike the strings against a kind of fretboard. I've never played one - or even heard one in person - but from what I've read the fingers can press against the hammer mechanism enough to slightly bend strings. I get the impression that touch on a clavichord is very different than a piano. It may be interesting to check out.
Dr Phil Brissenden, who lectures in Popular Music at Salford University, expresses very similar sentiment to you about the pros and cons of piano and guitar playing and has dedicated a considerable portion of his life to the invention of an instrument that bridges the gap between the two. The resulting creation, while not fretted, is a great deal like what you describe! It is a lap instrument, based on the body of an autoharp, but has a reverse damping system that uses a piano keyboard. It uses guitar-like strumming patterns in the right hand (currently based on flamenco guitar techniques) while the left hand controls chord patterns on the keyboard.
Here's his website:
There's also a Facebook page.
The chapman stick is not played on the lap, but otherwise fits most of your criteria. It's a 10 or 12 string guitar-like instrument, usually played by double-handed tapping.
While not held in the lap, take a look at the koto - Japan's national instrument.
Every string has a separate bridge which can be freely moved for tuning. I think I've seen an ensemble where part of the musicians moved the bridges during pieces, but I'm not sure about it.
Check out the Chinese guqin:
It can be played on table or lap: http://www.silkqin.com/04qart/qinplay.htm
I love the harmonics and barely audible string slides employed in its music.
yazh is an instrument which is vertical standing 1 or 2 or even more players can play that single instrument
it is quite big for 3 player standing as well as available smaller fits a lap
actually the same instrument is horizontally placed inside a grand piano
yazh has many varieties and many sizes and many strings, it is one of the oldest instrument of human race
I think it fits your description, it is called as
Harp in english, in Tamil we call it as
There is such an instrument which is called ‘kanklės’. This instrument is Lithuanian National instrument and has 29 strings and levers as well it is very similar to a harp. You can put it on your lap but we also use a stand for it. We change tonalities with the levers on the side. Originally, it is a folk instrument but musicians developed it and we started playing pop, classical and other music. Check the videos below: