I know you've already gotten a lot of suggestions, and I'm going to have to agree with Betty O. (who also has an early childhood educators background...don't think she mentioned it, but she knows thereof she speaks.) As a mom/former nanny/preschool teachers I have experienced a lot of problems around situations like this.
I noticed many parents cautioning that you should impress the 'value' of this object upon your child. Uphill battle, that. Three year olds do not have the ability to cognitively understand 'value' in the way that we adults do, and this sort of responsibility isn't appropriate to your son's age of development.
If you want to teach 'taking care of' an object, a child-friendly CD player with NO headphones is recommended. CD players are bigger, harder to lose, are less likely to be washed, dunked into a toilet, or squirreled away in a small place. As a parent or adult-in-charge, I want to hear what the kids are listening to. You just never know. (It's the same reason we feel that the kid's computer is better-kept in the kitchen or common areas, where *everyone* can see it, than in a bedroom, right?:))
Or even better, if his mom has a cd player, just make him a couple cds. And let her use her own discretion at what she wants to hear. Your son may have special music he likes at her house too. This is part of the give and take that coparenting with another person requires.
Perhaps his mom *hates* Irish music? MIght be a reason to consider this, but still..My father had to deal with those very real situations when my folks divorced. The result was that we had some special music and toys at dad's house and some stuff that was more relevant to mom at her house. Kids adapt. If this is something special he shares with you, do not force his mom to facilitate it.
I have also noticed children at this age becoming very directive about listening to music. They sometimes want to hear the same song...again and again and again. While the easy answer would be to allow headphones, there are many great reasons for waiting on those. One, your son doesn't have the capacity to understand 'safe' volume control. I'd also be wary of having him off in his own little world where he cannot hear what's happening around him as well as he should. If you think a kid can tune you out with 'selective hearing', just wait until they have limited their hearing to the few millimeters beyond their ears. This just teaches disconnection, which is turning into a sad lifelong habit, if you have read studies on children and media. Scary.
I liked the idea of making a memory for a gift and taking him to see music. See if you can make a video of your going together (Where your son is the star instead of the music) or take lots of pictures and make a book. Far more meaningful and he can recall the story of that time, this good time he had with you.
One last thing: if you give a child headphones and a listening device, very sophisticated for his age, what is there for later? I was 8 when I received an a.m. transistor radio with an earphone (a single one, not both ears) and loved it. It would have been sad if I'd been 'gadgeted out' at a young age--the radio probably wouldn't have been as valuable to me. I took very good care of that radio and when my younger brother was old enough, passed it down to him. (I was given a Radio Shack stereo at 12 or so, and this fit the bill for my age.) The point is, you have plenty of time in his life to give him fancy things--right now, the most important thing you can give him is time with you.