Lab Grown, or "cultured diamonds", are a bit more common now in stores. If you ask, they'll let you know which brands are man-made. I think on average they'll cost you between half and 1/3rd less than a natural diamond (based on the stone alone). Chemically they are the same and they are both diamonds - it's compressed carbon, the only difference is one was made in a machine and one was mined by some poor sod in a 3rd world country.
Diamonds aren't intrinsically as valuable as their price lets on. The small diamonds you find in most rings are pretty common, but the diamond cartels have an iron grip over the market and can effectively create false scarcity, which when combined with effective advertising over the past 70+ years, lets them command high prices. The only truly valuable diamonds are the naturally occurring colored ones or the super large ones, since they are actually rare.
Of course, none of that may matter to your significant other and they may just want a diamond because "that is what is expected". But I suppose you could gain something from a non-diamond engagement ring depending on their personality and likes. Emeralds are legitimately more rare than diamonds and can be just as pricey, for instance, so if they liked green or had Irish ancestry or have a May birthday or are obsessed with the Wizard of Oz, it might go over better.
Although if they were a true nerd or budding paleontologist/Jurassic Park junkie and could actually appreciate it, then the rarest and unable to be duplicated option would be a piece of amber with a trapped life form from the dinosaur era. Unless someone builds a time machine, no machine is ever going to be able to replicate that. But it would probably take a pretty unique woman to love that lol