Thanks to @matt-ward input, I finally figured this out!
I used Process Monitor to find out where SharpDevelop was looking for the DLL.. Here's what I found...
C:\Users\ghigad\Projets\CS\TestT4\packages\System.Data.SQLite.Core.1.0.92.0\lib\net40\x86\SQLite.Interop.dll
C:\Users\ghigad\Projets\CS\TestT4\packages\System.Data.SQLite.Core.1.0.92.0\lib\net40\Win32\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\x86\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\Win32\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\ADDINS\MISC\TEXTTEMPLATING\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\x86\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\Win32\SQLite.Interop.dll
C:\Users\ghigad\Projets\CS\TestT4\packages\System.Data.SQLite.Core.1.0.92.0\lib\net40\x86\SQLite.Interop.dll
C:\Users\ghigad\Projets\CS\TestT4\packages\System.Data.SQLite.Core.1.0.92.0\lib\net40\Win32\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\x86\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\Win32\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\ADDINS\MISC\TEXTTEMPLATING\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\x86\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\AddIns\Misc\TextTemplating\Win32\SQLite.Interop.dll
C:\Users\ghigad\Projets\CS\TestT4\packages\System.Data.SQLite.Core.1.0.92.0\lib\net40\SQLite.Interop.dll
C:\Program Files (x86)\SharpDevelop\4.4\bin\SQLite.Interop.dll
C:\Windows\SYSWOW64\SQLite.Interop.dll
C:\Windows\system\SQLite.Interop.dll
C:\Windows\SQLite.Interop.dll
C:\Windows\SYSWOW64\SQLite.Interop.dll
C:\Program Files\Common Files\Microsoft Shared\Windows Live\SQLite.Interop.dll
C:\Program Files (x86)\COMMON FILES\MICROSOFT SHARED\Windows Live\SQLite.Interop.dll
C:\Program Files\Java\jdk1.6.0_30\bin\SQLite.Interop.dll
C:\Program Files\apache-ant-1.8.2\bin\SQLite.Interop.dll
C:\cygwin\bin\SQLite.Interop.dll
C:\Windows\SYSWOW64\SQLite.Interop.dll
C:\Windows\SQLite.Interop.dll
C:\Windows\SYSWOW64\wbem\SQLite.Interop.dll
C:\Windows\SYSWOW64\WindowsPowerShell\v1.0\SQLite.Interop.dll
C:\Program Files\Intel\WiFi\bin\SQLite.Interop.dll
C:\Program Files\Common Files\Intel\WirelessCommon\SQLite.Interop.dll
C:\Program Files (x86)\COMMON FILES\Roxio Shared\DLLShared\SQLite.Interop.dll
C:\Program Files (x86)\Common Files\Roxio Shared\OEM\DLLShared\SQLite.Interop.dll
C:\Program Files (x86)\Common Files\Roxio Shared\OEM\DLLShared\SQLite.Interop.dll
C:\Program Files (x86)\COMMON FILES\Roxio Shared\OEM\12.0\DLLShared\SQLite.Interop.dll
C:\Program Files (x86)\Roxio\OEM\AudioCore\SQLite.Interop.dll
C:\Program Files (x86)\MICROSOFT SQL SERVER\100\Tools\Binn\SQLite.Interop.dll
C:\Program Files\Microsoft SQL Server\100\Tools\Binn\SQLite.Interop.dll
C:\Program Files\Microsoft SQL Server\100\DTS\Binn\SQLite.Interop.dll
C:\Program Files (x86)\MICROSOFT SQL SERVER\100\Tools\Binn\VSShell\Common7\IDE\SQLite.Interop.dll
C:\Program Files (x86)\MICROSOFT SQL SERVER\100\DTS\Binn\SQLite.Interop.dll
C:\Program Files\TortoiseSVN\bin\SQLite.Interop.dll
C:\Program Files (x86)\QuickTime\QTSystem\SQLite.Interop.dll
C:\Program Files (x86)\Windows Live\Shared\SQLite.Interop.dll
So, it turns out that quite a few directories are searched, then the contents of my PATH variable is used to search the file...
Matt's idea of putting the dll in SharpDevelop's bin directory would have worked, but I don't like the idea of saving files in the directory of an app... It would mean that I would have to copy the files over again if I ever update SharpDevelop to a newer version...
From the log, I guess copying the dll from .../packages/content/net40/x86 to .../packages/lib/net40/x86 would have also done the trick... Same thing here... If I fool around SQLite's Nuget package directory structure, everytime I'll check out my code from the source control or upgrade to a newer version, I'll have to copy the file again...
I finally decided to copy the DLL in a directory in C:\Program File (x86) and added it to my PATH.
Works great!
Thank you Matt for putting me on the right track!