I would extend the concrete out to the edge of the L shaped piece. It looks like the area doesn't get enough light for grass.

Or maybe install a path, if needed, or if you do not want to, or cannot, extend the patio Does anyone ever walk off the patio and head towards a particular place through the back yard? It wouldn't even have to be a poured cement sidewalk - there are other options. Can you post another picture of the side looking towards the tree trunk that is at the right of the bottom picture?

A non-grass ground cover such as creeping thyme. It bears heavy foot traffic, stays low so needs no mowing, and requires less sun.

If going with this method, I would use a garden claw to break up the hard earth this fall, scatter thyme seed, and leave it over winter. Freezing improves the germination of most seeds and the spring rains will keep it moist enough as it establishes.

Is there a drainage problem? Which direction does the back of the house face and what time of day was the photo taken? Does the dirt area get any sun? It looks like it might. Have you battled grubs etc.? Do you do a 4 step system, aerate, etc. (guessing not).

I'd amend the soil and sod the dirt if it gets any light at all. Or do deep shade grass seed or hydro seed. Then maintain it (prob 15 min / month to throw down some kind of 4 step system etc. (look for organic and shady) and 20 min a week to mow.

(The quickest/easiest is to have a landscaper come in and do the whole thing for you then all you have to do is water. But it will cost a bit.)

Your patio seems too small for a table like that, but perfect for a little outdoor couch, a few comfy chairs and side tables for your lemonade, and a kid-friendly coffee table/activity table in the middle for the little guy. Think of the patio as an outdoor living room. It already has 3 walls.

You have shade and by the looks of it, at least one kid. The only way you'll have a harder time growing grass there is if you adopt a couple of bull mastiffs. Some means of extending walkable surface past that point (as others have suggested above) is the ticket.

you gotta do what the grass wants to do. the grass doesnt want to do what you are asking of it. and the grass will win the battle every time.

make a 'play lawn' in a sunny spot where it can have healthy growth, and for shady play, make a solider surface.

this is my parents house, and yes - it's completely true that grass won't grow there. the patio faces the lawn and the mud/dirt gets tracked in by the grandkids. I'ts also just such an eyesore (the patio included). I guess I wondered whether people thought a concrete extension of the patio, pathway with plantings and/or wood deck would look nicest.

my parents already have a covered patio with couches and tables adjacent to this outdoor area (not pictured), and this is where we dine outdoors while the kids play. I think the transition is the thing we're looking for the most efficient and cost effective solution to.

thanks everyone for your insight :)