hickima ?

I was thinking along the same lines as Janelle.  Maybe add a pinch of sugar, a pinch of dill (or pickling spice mix) and a bit of vinegar - it would be pretty easy to customize to your child's taste.  Or, you could even just pull a little out for your child before adding the relish to the rest of the batch and not add anything to replace it - my kids would have been fine with that.

It may be the area that I live in, but other types of pickled vegetables are pretty available here - I would probably try carrots, green beans or asparagus and just chop them finely to get the pieces to be relish sized.  If that isn't available you could try giardiniera (a mild variety, unless you/your child like spicier foods) chopped finely. 

Another option might be to boil your potatoes with some pickling spice in the water or marinate the tuna with pickling spice and a bit of oil.

I have a kid that is allergic to cucumbers. I'm looking for some good substitution for pickle relish in potato salad and tuna salad mostly.