Bypassing 10 character limit of field name in shapefiles?
Sorry, the answer is no. You have to roll out your own field mapper and only software that uses your mapper will understand it. You could use other formats that do not have this limitation though (e.g. file geodatabase, spatialite, etc).
Some word of advice about the workarounds from personal experience.
When people choose shapefiles (and insist on them) as their main format, it is usually chosen for interoperability - think of it as adhering to a spec. If you choose to roll your own field mapper, you are basically doing quite the opposite - since you are doing things outside of a spec - you have created your "extended spec".
Have I done this in the past? Yes. And it most certainly always turns to more of a pain than actually solving a problem because every time you try to open the shapefiles into anything else that can read/write shapefiles, you end up with a table with a whole bunch of hard to understand fields.
At that point, I would ask you, why are you using shapefiles? Either come up with a workflow solution that sticks with the shapefile spec and its limitations, or change file formats. Everything else is just a recipe for headaches.
There is a standard way to deal with this, although your clients might not be completely happy with it: you export two files, a shapefile and a data file in a format their software can read. The shapefile has only a unique identifier, [Id], for attributes. The data file has several attributes: [Id] to match the shape, [Field] to provide the field name, [Type] to indicate its type, and one attribute of each possible data type to store the value. Each field in the original file is stored as a record in this data file.
For example, a source table looking like this:
[Shape] [Id] [Name] [Population2010]
shape1 A1 California 37253956
shape2 A2 Texas 25145561
shape3 A3 Wyoming 563626
would have a corresponding data file
[Id] [Field] [Type] [Text] [Integer]
A1 Name Text California <Null>
A1 Population2010 Integer <Null> 37253956
A2 Name Text Texas <Null>
A2 Population2010 Integer <Null> 25145561
A3 Name Text Wyoming <Null>
A3 Population2010 Integer <Null> 563262
It should be obvious how to use these data in any RDBMS and how to convert back and forth between the two formats.
If your customer is using ArcGIS, you could provide a script to assign field aliases in bulk. This would give them the appearance of long field names when they're using the data.
Similar scripting may work to provide aliases in other GIS packages, too.