What are some options for online and offline particle tracking in ocean models?
The main difference between online and offline trajectory calculation is that the online is implemented as part of the model, and the trajectories are part of the model output. Offline trajectory calculation is a separate piece of software that takes Eulerian model output fields as input and calculates the trajectories.

The main advantage of online particle tracking is that the displacement of a particle or tracer can be evaluated every time step, no matter how small it is. As a result, you can obtain a true particle trajectory based on what model produces. The disadvantage of the online tracking is that the user needs to know the particle release location and time ahead of running the model. The models listed below (at least) support online Lagrangian trajectory calculation:


HYbrid Coordinate Ocean Model (HYCOM) is a hydrostatic, hybrid-coordinate ocean circulation model that has particle calculation built-in;
MIT General Circulation Model (MITgcm) is a non-hydrostatic circulation model with many physical parameterization options and a single dynamical kernel for both the atmosphere and ocean;
Regional Ocean Modeling System (ROMS) is a hydrostatic, sigma-coordinate (terrain-following) ocean model with many physical parameterization options a large user community.


On the other side, offline particle tracking can only use the Eulerian fields as often as they are output. Thus, there will be an error in the calculated trajectory depending on the frequency of the output. On the other side, the main advantage of offline calculation is that the user can experiment and play around with deployment locations, times, area of coverage etc., because trajectory calculation typically takes much less computational effort than the actual model. There are several options here:


Connectivity Modeling System (CMS) takes gridded NetCDF files as input and produces Lagrangian trajectory output. Can handle fields from various Arakawa-staggered grids, and may produce probabilistic trajectories, diffusivity specified by user (optional). Compatible with at least HYCOM and ROMS output.
Ariane also takes gridded NetCDF files and is compativle with at least HYCOM, ROMS and NEMO models.
TRACMASS seems to be compatible with many circulation models. It also has a Python interface.


All models and tools listed in this answer are freely available for download under different open source licenses.