L Pile is an industry standard software to assess lateral deflection of piles.

When we produce geotechnical interpretative reports we provide the lateral force that results in 12mm (half inch) deflection. This is taken as the allowable lateral capacity in my part (and other) parts of the world.

I recently received a very helpful LPILE anlysis from a geotechnical engineer. He sent me graphs for both free head and fixed head conditions for a typical size pile. I'll need the free head condition in this case.

One of the graphs shows depth vs. displacement for 1, 2, 3, 4, and 5,000 lb lateral load at the top of pile.

We're just getting started with design, so I don't have any preliminary loads. Once I figure out the total lateral load on the building, the graphs will let me determine pile placement and number of piles tuned to the lateral deflection limits I'd like to keep.

Depends on your local code. Here we have to make analysis of piles under lateral loading if soil profile is Site Class E (soft soil profile) or F.
If you dont have LPile, you can use Evans-Duncan method and calculate the lateral load that will result in 0.5 inch lateral deflection (y). You can find that in the Reese book "Single Piles and Pile Groups Under Lateral Loading".

Or you can take 1 inch, as recommended in IBC:

"1810.3.3.2 Allowable lateral load. Where required by the design, the lateral load capacity of a single deep foundation element or a group thereof shall be determined by an approved method of analysis or by lateral load tests to not less than twice the proposed design working load. The resulting allowable load shall not be more than one-half of the load that produces a gross lateral movement of 1 inch (25 mm) at the lower of the top of foundation element and the ground surface, unless it can be shown that the predicted lateral movement shall cause neither harmful distortion of, nor instability in, the structure, nor cause any element to be loaded beyond its capacity."

I would caution the use of 1" lateral deflection as the limiting criterion. Depending on the structural system, 1" of lateral movement can induce significant connection and member stresses. If the structural system design is marginal or "highly optimized", localized failures can occur.