We love the name [name_m]Lincoln[/name_m], however we plan to call him [name_m]Link[/name_m]. So which of these look better?
[name_m]Lincoln[/name_m] n.n [name_m]Linc[/name_m]
Linkon n.n [name_m]Link[/name_m]
Linkoln n.n [name_m]Link[/name_m]
[name_f]My[/name_f] partner really likes [name_m]Lincoln[/name_m] with the nickname [name_m]Link[/name_m]. I really don’t think you need to change the spelling to get the nickname [name_m]Link[/name_m].
Please stick with [name_m]Lincoln[/name_m]. 
The traditional spelling is so handsome. I would call him [name_m]Linc[/name_m], but I see no reason you couldn’t use [name_m]Link[/name_m] if you prefer the K.
[name_m]Lincoln[/name_m] looks the best. The other two look silly. There’s no reason you can’t spell his nickname [name_m]Link[/name_m] and put [name_m]Lincoln[/name_m] on the birth certificate.
Well, no matter how you spell it, other people will want to spell it [name_m]Lincoln[/name_m]. Because that’s how it’s spelt. So as I see it, you really might as well do your son the favour of not deliberately causing him hassle in his life. If his name is Linkoln or Linkon, he’ll have to correct people all the time, “No, it’s with a k [and no l]”. And then people will ask why, and maybe think his parents didn’t know how to spell [name_m]Lincoln[/name_m].
Choose the path of least resistance. There’s no reason why his nickname can’t be [name_m]Link[/name_m]. A lot of people are named [name_m]Thomas[/name_m] and called [name_m]Tom[/name_m], for example, where’s the problem there?
You do not need to change the spelling to get the nickname at all. I would spell it the traditional way: [name_m]Lincoln[/name_m]. It looks much nicer, more classy, etc.