I was hoping that someone in the Skokie-Evanston area of Chicago had a clipping of a lemon meyer tree. Will come and pick it up.
intohouse, citrus don't grow so well from cuttings. Not that you couldn't try, but you'd be better off grafting the cutting to rootstock. Or, simply buying a grafted Improved Meyer lemon tree (if budget permits).
Also, the Improved Meyer is a bush, not a tree. If you get a clipping to grow, it will not naturally grow as a tree. Like oleanders, pruning just encourages the Meyer to send out new sprouts and become an even thicker bush.
Improved Meyer is one of the few citrus that can be found at nurseries as rooted cuttings, although the majority of meyers offered here (Tucson) are grafted. The reasons meyers are offered as cuttings are because they do pretty well on their own roots, they root fairly easily, and if frozen to the ground (occasional event here in Tucson, AZ) they can (with luck) resprout true from the trunk and roots. Meyers are also one of the few citrus that aren't true from seed, so if you want a clone it has to be a graft or a cutting.
I've rooted probably a dozen varieties of citrus, with varying results. Sweet and sour oranges are fairly easy and do well. Likewise grapefruit and lemons are relatively easy and most varieties do okay on their own roots. Mandarins are difficult to root but usually do okay on their roots. Trifoliate rootsocks are very difficult to root.
Most citrus will come true from seed so rooting cuttings is not necessary if you want a "clone" of most varieties on their own roots. Many gardeners feel that seedlings have superior roots to cuttings, although that isn't true in all cases from my experience. I have had good success with sour orange cuttings as rootstocks. Flame grapefruit and meyer lemon cuttings have done so-so as inground plants. Generally it is well worth the effort to get trees grafted onto the best roots for the particular situation though. Exceptions being the odd case of lemons & limes planted in-ground in marginal winter climes where they occasionally freeze to the ground.
Good luck with rooting.
Actually, it's a common myth that Meyers don't come true from seed. What I think is correct is that Meyers have a REALLY low viability from seed...and I do mean, really, really low.
My neighbor had a Meyer but never picked them (she never picks any of the fruit from her different trees - go figure!). It was on the border of her property and ours. Dozens, maybe hundreds, fell off to rot on the ground the first five years we lived here.
Then one single plant took root. One, out of all those fallen Meyers! It is in the worst possible place you would pick: a dead-end narrow aisle with greasy adobe clay soil that slides downhill, northern exposure, permanently shaded by a 12' tall shed right in front of it.
That was 15 years ago. It is now 8' tall and at least 10' across, and bears Meyers almost year-round. You can search on my name and in one of the discussion threads here, I've posted photos of my three Meyers, including this one, from earlier this year.
If you are going to grow the Meyer in a container, as I am sure you will in Northern climes, a rooted cutting is a reasonable choice; it is a true clone, thus it is older and will bear fruit faster. I think you need a true greenhouse environment to get a cutting to root; but it is a common way of propagation in the North. If your were going to put it in the ground, I would advise against a rooted cutting, as the Meyer root is not very aggressive, resistant, or fast growing.