You are certainly free to develop a more meaningful relationship with your familiar. In fact I've always handled my familiars as a distinct entity, a friend and companion with its own quirks, not just some summoned spirit.
In regards to having it scale and grow, I'll need to think on that, and get back to you on it. At minimum you'd need to give up something in exchange for any notable amount of power gained.
Edited: daishain on 21st Dec, 2017 - 7:18pm
That's the tradeoff in 5e. They scaled much better in 3.5 but it was a lot more than a few gold pieces to summon them again. What of the warlock's improved familiar? They should retain an advantage over a regular familiar.
I haven't earned any gems here on the site, Gknight, and am thus no expert but perhaps a crystal could be turned in for a boost to the familiar? I know that's how Ky got Mari earlier in the campaign as a level3 side character, I believe. Perhaps you have to take what the random gem effect is though, I'm not sure.