I have a relative who I don’t want to cut off, but it’s obvious they need me more than I need them and it’s exhausting. They want me to drop everything and spend time with them at a moment’s notice, usually eating out which is super expensive, and they constantly have mental health crises and text me that they “need” me.
I can say no to them, that’s not the problem, but they haven’t gotten the hint and are just as clingy years after years of it. It’s really unhealthy for me to be their only friend when I don’t feel the same way. Is there any way to encourage them to expand their social life without sounding like a dick? I have no social grace and sound like a dick a lot.
They probably know a lot of this.
They may not know it’s a problem, or if they do know, then they probably don’t know what to do about it.
So that’s how I’d go about this. Rather than saying they need to expand their support network, I’d ask them what stops them from expanding their range of friends, support and opportunities for connection?
Means you’re not pressuring them or saying you know what they need better than they do. It’s asking about them, what they think and feel.
Bingo.
It is likely that they have burned friends out one by one and all they have left is you. BUT, you have to have your boundaries.
It sounds like they need therapy and possibly medication, if I am reading between the lines here. That would be the first step to changing the pattern for them. Unfortunately, all you can do is recommend things. They have to take the next steps. 
I agree that therapy would be valuable here.
There’s stuff it helps with and stuff it doesn’t. Learning how to relate to others and working through what stops you is one of those things it’s really good for.
Therapy is great if you’re lonely because the only thing worse than only having one person to unload your thoughts onto is having zero people. And who better than someone who you pay to be a thoughts unloading person.
This sounds like a good way to approach it, thank you.