What are "buy-in rights"?

In the IP context, legally speaking it is "controlled transfer of interest in pre-existing IP, in exchange for arm’s length consideration (buy-in payment)". Basically in a multi-party deal, or IP-pool, you want late-comers to compensate the founder/core IP for sunk R&D, since this may involve revaluing the "property", it may trigger tax consequences (xref transfer pricing, especially in international context).

In investing context, it may simply be an option to purchase shares at a prior agreed price, but there are nuances so more details needed to correctly answer.