To tell the truth, I would hand her my phone number/email address and say "we'd love to have __ over to play sometime, or to meet up at a park" and leave it at that. She's made excuses several times, I'd not keep bringing it up. You don't really know what the reason is, and it may have nothing at all to do with you or your child. They may have a home situation that you're not aware of, and it really wouldn't be your business if she doesn't choose to divulge it to you. (Not saying anything "serious", nor do I mean that to sound mean, but simply that there might be other factors...and she may not feel the desire or need to share those). Like another mom said, we all have our right to privacy.
The children are in kindergarten, which has a lot of play time. I say after handing contact info over to the friend's mom, drop it and let the children play AT SCHOOL. When they're a little older, if they're still friends and have similar interests, perhaps they could have an enrichment activity together (Tball, scouts, etc).
It isn't "hurtful" to tell the truth. I'd just tell my son, "Big man, you saw and heard me invite __ for a playdate. I gave his mom my phone number, and it's in her court. If she doesn't let him play, there's really nothing we can do about it, but you guys have fun at school together ok? Don't make your friend feel badly if he can't play, just have fun when you do see him. Maybe next year y'all can join cubscouts, or play soccer together. And of course we'll ask him to come to your birthday party". The end. Nothing else needs to be said about it because everything else would just be guessing and conjecture anyway.
As a kid, I had a friend that was a different religion and not really allowed to hang out with kids outside her church, and school was her one way to play and talk with other kids. As a young adult, I had a friend whose very traditional, strict Indian family was very hesitant to let her hang out with families outside of "the book". As an adult, a neighbor with a child 1 year older than mine would call me to come hang out, but if someone else came by she would really act like I had just showed up and she didn't know why...embarassed that I was there. I thought that was sad for her---she was very insecure and working too hard to be "the Keller mom". (My husband and I used to have a little laugh about how the clique she tried to stay with all drove the same car, all wore the same clothes (the trendy, casual but not really stuff--where they spend 45 minutes to look like they just got up and stepped out for the newspaper but had to get the look "just so"), all went to yoga and coffee afterwards (which cracks me up---counter intuitive in my opinion, to put the 2 together), etc, etc. Basically, I didn't fit the bill. I didn't wear the right clothes, look the same way, or do the same things, and that is ok. I was fine with who I am though, and didn't "play" with her after 2 or 3 times of that happening. Another mom who I've known since college married a guy who is just a real jerk. Not "abusive" in the typical sense, but he had the stupidest rules! He wanted the kids to hang out only with people he knew, the mom needed to come home and do a little list or chores, homework, have dinner on the table when he got home, and then drop everything when he was home for manditory "family time". They had to watch a litany of football, baseball, and nascar....nothing exciting, but it was what he wanted of his family. I would slit my wrists or his throat if I was married to him, but it worked for them. The kids were fine and cared for, they loved their dad and loved watching sports. But they don't go out really, at all. She may have a sick mom to care for. Or a job to get ready for. Or be super shy or have mild anxiety in social situations. Or just not LIKE the whole playdate thing. If you're my age, you'll remember that we N E V E R had "playdates". Kids just met up and played in their own neighborhoods. You saw the kids that weren't in your neighborhood at school or other activities.
It could be ANYTHING. I wouldn't sweat it at all. When she has your contact info, you've taken the step to befriend her and that is all you should do. If she's interested or able, she will take the next step.