My mother's latest boyfriend - call him Grandpa?

Just calling someone "Grandpa" doesn't give them special privileges. Just make sure that you explain it all to your kids. You don't know him well, and you don't know how trustworthy he is. He may never earn your trust. But just to keep the peace, call him "Grandpa." Assuming that he wants to be called that. 

As a divorced grandmother who has post-divorce relationships I would not want my grandkids to call my fiance "Grandfather", or Abuelo, or Papi (spanish for grandpa). Even yestereday my son was explaining to the five year old twins that their grandfather (dec) loved them very much. Some day they will understand the family tree. Mileage may vary.

I would compromise and call him Grandpa Joe (or whatever his name is). You kids will hear everyone calling him Joe and will probably call him Joe most of the time too except when you say "please call him Grandpa Joe when Grandma is around because it makes her happy". Then if your mother starts dating someone else, that person can be Grandpa Mike or whatever. Your kids will not be more or less attached to the person because of a name so I wouldn't worry about that. I would worry more about your kids getting very attached to him and them they break up, but that is a different question. You could also explain to your child that this is Grandma's boyfriend and they are not married and he is not a blood relation and explain the difference between their blood grandpa, your father, who will always be their grandpa no matter what, and this person who Grandma is dating who is much newer to your family (and by implication more tangential). Kids eventually get it. We have a similar situation and around 7 my kid started to understand that his blood grandpa died before he was born but he gets some of his genes from him and so do his cousins, but this other person who is newer to all of us (married to grandma in our case) is not a blood relation but we still call him Grandpa Joe.  

Family is important.  I feel as though the transient nature of your mother’s relationships has no benefit to your kids. I wouldn’t ask my kids to call anyone something just to appease someone else.  You said so yourself - the relationship is not likely to last.   Why give someone an important title in the kids’ lives knowing that the person isn’t likely to be around?  Call him by his name.  Think of peace in the long term... 

My grandma's second husband was "grandpa," but we last had contact with him when I was in preschool. It is a little strange that he was the only "grandpa" I ever knew, but he was not a big part of my life, but it was not a big deal.

My kids call their cousin's grandpa "grandpa" even though they know he's not their grandpa. Think of "grandpa" as an honorific for an older man. Call him "Grandpa John" instead of just "grandpa," and it sounds a bit less intimate.

The situation you describe is a tricky one, with many viewpoints to be respected and emotions to be addressed. You didn't mention whether your father is alive and in the picture. I mention this because my mother-in-law wanted my children to call her husband (married late in life) by a familial label. I felt this would be very confusing to my children, and my husband was offended as he felt it disrespected his father. Yet, I understood that my mother-in-law did not want the children calling her husband by his first name.

You might consider having a conversation with your mother, asking what is important to her about the word "Grandpa" and what that might mean for her and her boyfriend. Share your concerns about using the term. I suggest putting aside anything that might seem judgmental, such as potential longevity of the relationship. Ask if she will engage with you in brainstorming other potential terms and see if any of them seem to stick. In our case, Papa (name), has worked. Although, now that my children are into their late teenage years, they bristle at even that, reverting to my mother-in-law's husband's first name or nothing at all, even though they each have a good relationship with him.

Good luck navigating these tricky waters.

To make it more informal, you could try adding the man's name to the title, i.e. "Grandpa Mike"  rather than just "Grandpa."  To me, this takes some of the power out of the title.  You could also try to "make up" a term for him that isn't actually Grandpa, but has some of the same connotations, such as "Pops" or (as in the example above) "GM" (for "Grandpa Mike").  In several cultures, terms like Auntie or Uncle are given to lots of people who are not actually related to a child.  On the other hand, if you feel strongly about this, I think you are well within limits to say that he doesn't get called Grandpa until they are married (or maybe not even then, depending on how you feel about it).  

I don't think your impulsive mother should encourage the kids to call her boyfriend "Grandpa" without consulting you first. You don't have to construct a "valid" concern; they're your kids, and you don't know him very well (and I'm not imagining him as a pedophile, either). Children can get attached real fast and it sounds like your mother breaks up with guys that way, too. It would be sad for them to suddenly lose a grandpa figure, so I wouldn't present him as one until he and she are together for a few years.Tell her, if she demands an explanation, just what you've told us. For now he could be "Grandma's friend Horatio"; the grandpa word should be your decision, and your children's, not hers.