I’ll definitely add that to the list! Thank you!
Her maiden name is [name_m]Lillard[/name_m], so that’s not something I feel like we can work with either.
You could use Marces (smoosh of [name_f]Mary[/name_f] [name_f]Frances[/name_f]). I saw it listed on another site. Marcesa.
My friend followed a naming tradition in her background where you just use the first syllable or sounds in a name to honor a relative (they don’t reuse names across generations in her family). So [name_f]Frances[/name_f] would turn to [name_f]Frida[/name_f], for example. I thought that was nice because it gives the parents a bit more freedom while still giving each child’s name a link to a relative.
Lots of good names have already been suggested, I only thought of one to add:
[name_f]Saffron[/name_f]
Maybe it’s too much of a stretch, but it reminds me of the two syllables of [name_f]Frances[/name_f] in reverse order!
I also think you could take any [name_f]Lilly[/name_f]/Lily name from [name_m]Lillard[/name_m]!
Or you could also try to get the same initials, with or without the [name_f]Mary[/name_f] part. I also think you could consider a name that starts with “[name_f]Mar[/name_f]” besides [name_f]Mary[/name_f] and that could be a special connection between your grandma, yourself, and your daughter.
[name_f]Marceline[/name_f] [name_f]Frida[/name_f] [name_f]Lily[/name_f]
[name_f]Fiona[/name_f] [name_f]Lillian[/name_f]
Etc.
I just thought if Françoise as well—maybe a no go since is also starts with “[name_u]Fran[/name_u]” but if either or you are Francophiles, it could work 
Francesca is a beautiful alternate, IMO.
[name_f]Francine[/name_f] may be another option.
Hmm but she dislikes it in all its forms, I see.
Okay.
Howabout… Instead of naming baby after her, name baby something reminiscent of an activity or book you and your grandmother enjoyed doing together… And will always remind you of her.
I think using the initials of your grandmother would be a subtle way to honor her. MFL
Here are some suggestions:
[name_f]Margot[/name_f] [name_f]Fianna[/name_f] (or [name_f]Fiona[/name_f]) [name_u]Leigh[/name_u]
[name_f]Margaret[/name_f] [name_f]Felicity[/name_f] [name_u]Laine[/name_u]
[name_f]Marcella[/name_f] [name_f]Faye[/name_f] [name_u]Louise[/name_u]
[name_f]Marielle[/name_f] [name_f]Faith[/name_f] [name_f]Leora[/name_f]
Francine or Francoise
Frankie 
Francesca
[name_f]Cecilia[/name_f]/Cecily
[name_f]Anya[/name_f]/Anna/Ana and variants
[name_f]Frida[/name_f]
[name_f]Freya[/name_f]
[name_u]Frankie[/name_u]
[name_f]Nancy[/name_f]
[name_f]Raina[/name_f]
You could also use her middle if she prefers that.
So I also would never want any grandchildren/great grandchildren named after me. In general I like my name but I don’t fee I need the honor and I like so many other names more than mine. If one of my grandchildren wanted to honor me I would prefer they give me a list of names they loved and let me chose the baby’s name from it. Maybe something to consider? I think anyone would feel honored to be involved in getting to choose a name for their great grandchild.
I like [name_f]Francesca[/name_f]. It feels more modern than [name_f]Frances[/name_f], while also sharing the nicknames.