James I wanted to be seen as a peacemaker and married his children to spouses of different denominations. What if he married his son to a protestant and his daughter to a Catholic instead of the other way around? Who would be Elizabeth's Catholic groom? I've heard that Anne of Denmark wanted a match with Savoy, but it seems like Savoy would prefer a French match. the Duke of Lorraine had to marry his cousin to consolidate his claim. The Hapsburgs were stuck in their inbreeding loop. Who would be her best/ most realistic option?
A match with Savoy was only on the table as long as Felipe III of Spain had no children/sons. Savoy WOULDN'T prefer a French match until just before Henri IV's death.
And there WAS talk of Elizabeth marrying Louis XIII under Henri's anti-Habsburg network.
The Habsburgs were not "stuck in their inbreeding loop", since both Matthias and Leopold V considered different matches to the ones they made (Matthias to Magdalene of Bavaria - mother of Philipp Wilhelm of Neuburg; and Leopold to first Feodosia Feodorovna then Xenia Godunova of Russia). Neither was overly well thought of by his family due to his "backstabbing" nature. So a match to Matthias doesn't seem out of left field. If only for the irony of Liz being LEGITIMATE queen of Bohemia.
Sigismund III of Sweden and Poland was interested in marry Arbella Stewart, but I think Liz might be TOO young to take advantage of that offer. Wladyslaw IV was interested in marrying Liz's namesake daughter, and Gustaf Adolf was likewise interested in Liz. But the Wasas might be a bit of a stretch